2引数が同じ3引数の論理演算子

2引数が同じ3引数の論理演算子
\(P,Q\)は命題変数とする。

肯定と肯定パターン

(1)

\[ P\lor\left(P\lor Q\right)\Leftrightarrow P\lor Q \]

(2)

\[ P\lor\left(P\land Q\right)\Leftrightarrow P \]

(3)

\[ P\lor\left(P\rightarrow Q\right)\Leftrightarrow1 \]

(4)

\[ P\lor\left(P\leftarrow Q\right)\Leftrightarrow P\leftarrow Q \]

(5)

\[ P\lor\left(P\leftrightarrow Q\right)\Leftrightarrow P\leftarrow Q \]

(6)

\[ P\lor\left(P\downarrow Q\right)\Leftrightarrow P\leftarrow Q \]

(7)

\[ P\lor\left(P\uparrow Q\right)\Leftrightarrow1 \]

(8)

\[ P\lor\left(P\nrightarrow Q\right)\Leftrightarrow P \]

(9)

\[ P\lor\left(P\nleftarrow Q\right)\Leftrightarrow P\lor Q \]

(10)

\[ P\lor\left(P\nleftrightarrow Q\right)\Leftrightarrow P\lor Q \]

(11)

\[ P\land\left(P\lor Q\right)\Leftrightarrow P \]

(12)

\[ P\land\left(P\land Q\right)\Leftrightarrow P\land Q \]

(13)

\[ P\land\left(P\rightarrow Q\right)\Leftrightarrow P\land Q \]

(14)

\[ P\land\left(P\leftarrow Q\right)\Leftrightarrow P \]

(15)

\[ P\land\left(P\leftrightarrow Q\right)\Leftrightarrow P\land Q \]

(16)

\[ P\land\left(P\downarrow Q\right)\Leftrightarrow0 \]

(17)

\[ P\land\left(P\uparrow Q\right)\Leftrightarrow P\nrightarrow Q \]

(18)

\[ P\land\left(P\nrightarrow Q\right)\Leftrightarrow P\nrightarrow Q \]

(19)

\[ P\land\left(P\nleftarrow Q\right)\Leftrightarrow0 \]

(20)

\[ P\land\left(P\nleftrightarrow Q\right)\Leftrightarrow P\nrightarrow Q \]

(21)

\[ P\rightarrow\left(P\lor Q\right)\Leftrightarrow1 \]

(22)

\[ P\rightarrow\left(P\land Q\right)\Leftrightarrow P\rightarrow Q \]

(23)

\[ P\rightarrow\left(P\rightarrow Q\right)\Leftrightarrow P\rightarrow Q \]

(24)

\[ P\rightarrow\left(P\leftarrow Q\right)\Leftrightarrow1 \]

(25)

\[ P\rightarrow\left(P\leftrightarrow Q\right)\Leftrightarrow P\rightarrow Q \]

(26)

\[ P\rightarrow\left(P\downarrow Q\right)\Leftrightarrow\lnot P \]

(27)

\[ P\rightarrow\left(P\uparrow Q\right)\Leftrightarrow P\uparrow Q \]

(28)

\[ P\rightarrow\left(P\nrightarrow Q\right)\Leftrightarrow P\uparrow Q \]

(29)

\[ P\rightarrow\left(P\nleftarrow Q\right)\Leftrightarrow\lnot P \]

(30)

\[ P\rightarrow\left(P\nleftrightarrow Q\right)\Leftrightarrow P\uparrow Q \]

(31)

\[ P\leftarrow\left(P\lor Q\right)\Leftrightarrow P\leftarrow Q \]

(32)

\[ P\leftarrow\left(P\land Q\right)\Leftrightarrow1 \]

(33)

\[ P\leftarrow\left(P\rightarrow Q\right)\Leftrightarrow P \]

(34)

\[ P\leftarrow\left(P\leftarrow Q\right)\Leftrightarrow P\lor Q \]

(35)

\[ P\leftarrow\left(P\leftrightarrow Q\right)\Leftrightarrow P\lor Q \]

(36)

\[ P\leftarrow\left(P\downarrow Q\right)\Leftrightarrow P\lor Q \]

(37)

\[ P\leftarrow\left(P\uparrow Q\right)\Leftrightarrow P \]

(38)

\[ P\leftarrow\left(P\nrightarrow Q\right)\Leftrightarrow1 \]

(39)

\[ P\leftarrow\left(P\nleftarrow Q\right)\Leftrightarrow P\leftarrow Q \]

(40)

\[ P\leftarrow\left(P\nleftrightarrow Q\right)\Leftrightarrow P\leftarrow Q \]

(41)

\[ P\leftrightarrow\left(P\lor Q\right)\Leftrightarrow P\leftarrow Q \]

(42)

\[ P\leftrightarrow\left(P\land Q\right)\Leftrightarrow P\rightarrow Q \]

(43)

\[ P\leftrightarrow\left(P\rightarrow Q\right)\Leftrightarrow P\land Q \]

(44)

\[ P\leftrightarrow\left(P\leftarrow Q\right)\Leftrightarrow P\lor Q \]

(45)

\[ P\leftrightarrow\left(P\leftrightarrow Q\right)\Leftrightarrow Q \]

(46)

\[ P\leftrightarrow\left(P\downarrow Q\right)\Leftrightarrow P\nleftarrow Q \]

(47)

\[ P\leftrightarrow\left(P\uparrow Q\right)\Leftrightarrow P\nrightarrow Q \]

(48)

\[ P\leftrightarrow\left(P\nrightarrow Q\right)\Leftrightarrow P\uparrow Q \]

(49)

\[ P\leftrightarrow\left(P\nleftarrow Q\right)\Leftrightarrow P\downarrow Q \]

(50)

\[ P\leftrightarrow\left(P\nleftrightarrow Q\right)\Leftrightarrow\lnot Q \]

(51)

\[ P\downarrow\left(P\lor Q\right)\Leftrightarrow P\downarrow Q \]

(52)

\[ P\downarrow\left(P\land Q\right)\Leftrightarrow\lnot P \]

(53)

\[ P\downarrow\left(P\rightarrow Q\right)\Leftrightarrow0 \]

(54)

\[ P\downarrow\left(P\leftarrow Q\right)\Leftrightarrow P\nleftarrow Q \]

(55)

\[ P\downarrow\left(P\leftrightarrow Q\right)\Leftrightarrow P\nleftarrow Q \]

(56)

\[ P\downarrow\left(P\downarrow Q\right)\Leftrightarrow P\nleftarrow Q \]

(57)

\[ P\downarrow\left(P\uparrow Q\right)\Leftrightarrow0 \]

(58)

\[ P\downarrow\left(P\nrightarrow Q\right)\Leftrightarrow\lnot P \]

(59)

\[ P\downarrow\left(P\nleftarrow Q\right)\Leftrightarrow P\downarrow Q \]

(60)

\[ P\downarrow\left(P\nleftrightarrow Q\right)\Leftrightarrow P\downarrow Q \]

(61)

\[ P\uparrow\left(P\lor Q\right)\Leftrightarrow\lnot P \]

(62)

\[ P\uparrow\left(P\land Q\right)\Leftrightarrow P\uparrow Q \]

(63)

\[ P\uparrow\left(P\rightarrow Q\right)\Leftrightarrow P\uparrow Q \]

(64)

\[ P\uparrow\left(P\leftarrow Q\right)\Leftrightarrow\lnot P \]

(65)

\[ P\uparrow\left(P\leftrightarrow Q\right)\Leftrightarrow P\uparrow Q \]

(66)

\[ P\uparrow\left(P\downarrow Q\right)\Leftrightarrow1 \]

(67)

\[ P\uparrow\left(P\uparrow Q\right)\Leftrightarrow P\rightarrow Q \]

(68)

\[ P\uparrow\left(P\nrightarrow Q\right)\Leftrightarrow P\rightarrow Q \]

(69)

\[ P\uparrow\left(P\nleftarrow Q\right)\Leftrightarrow1 \]

(70)

\[ P\uparrow\left(P\nleftrightarrow Q\right)\Leftrightarrow P\rightarrow Q \]

(71)

\[ P\nrightarrow\left(P\lor Q\right)\Leftrightarrow0 \]

(72)

\[ P\nrightarrow\left(P\land Q\right)\Leftrightarrow P\nrightarrow Q \]

(73)

\[ P\nrightarrow\left(P\rightarrow Q\right)\Leftrightarrow P\nrightarrow Q \]

(74)

\[ P\nrightarrow\left(P\leftarrow Q\right)\Leftrightarrow0 \]

(75)

\[ P\nrightarrow\left(P\leftrightarrow Q\right)\Leftrightarrow P\nrightarrow Q \]

(76)

\[ P\nrightarrow\left(P\downarrow Q\right)\Leftrightarrow P \]

(77)

\[ P\nrightarrow\left(P\uparrow Q\right)\Leftrightarrow P\land Q \]

(78)

\[ P\nrightarrow\left(P\nrightarrow Q\right)\Leftrightarrow P\land Q \]

(79)

\[ P\nrightarrow\left(P\nleftarrow Q\right)\Leftrightarrow P \]

(80)

\[ P\nrightarrow\left(P\nleftrightarrow Q\right)\Leftrightarrow P\land Q \]

(81)

\[ P\nleftarrow\left(P\lor Q\right)\Leftrightarrow P\nleftarrow Q \]

(82)

\[ P\nleftarrow\left(P\land Q\right)\Leftrightarrow0 \]

(83)

\[ P\nleftarrow\left(P\rightarrow Q\right)\Leftrightarrow\lnot P \]

(84)

\[ P\nleftarrow\left(P\leftarrow Q\right)\Leftrightarrow P\downarrow Q \]

(85)

\[ P\nleftarrow\left(P\leftrightarrow Q\right)\Leftrightarrow P\downarrow Q \]

(86)

\[ P\nleftarrow\left(P\downarrow Q\right)\Leftrightarrow P\downarrow Q \]

(87)

\[ P\nleftarrow\left(P\uparrow Q\right)\Leftrightarrow\lnot P \]

(88)

\[ P\nleftarrow\left(P\nrightarrow Q\right)\Leftrightarrow0 \]

(89)

\[ P\nleftarrow\left(P\nleftarrow Q\right)\Leftrightarrow P\nleftarrow Q \]

(90)

\[ P\nleftarrow\left(P\nleftrightarrow Q\right)\Leftrightarrow P\nleftarrow Q \]

(91)

\[ P\nleftrightarrow\left(P\lor Q\right)\Leftrightarrow P\nleftarrow Q \]

(92)

\[ P\nleftrightarrow\left(P\land Q\right)\Leftrightarrow P\nrightarrow Q \]

(93)

\[ P\nleftrightarrow\left(P\rightarrow Q\right)\Leftrightarrow P\uparrow Q \]

(94)

\[ P\nleftrightarrow\left(P\leftarrow Q\right)\Leftrightarrow P\downarrow Q \]

(95)

\[ P\nleftrightarrow\left(P\leftrightarrow Q\right)\Leftrightarrow\lnot Q \]

(96)

\[ P\nleftrightarrow\left(P\downarrow Q\right)\Leftrightarrow P\leftarrow Q \]

(97)

\[ P\nleftrightarrow\left(P\uparrow Q\right)\Leftrightarrow P\rightarrow Q \]

(98)

\[ P\nleftrightarrow\left(P\nrightarrow Q\right)\Leftrightarrow P\land Q \]

(99)

\[ P\nleftrightarrow\left(P\nleftarrow Q\right)\Leftrightarrow P\lor Q \]

(100)

\[ P\nleftrightarrow\left(P\nleftrightarrow Q\right)\Leftrightarrow Q \]

肯定と否定パターン

(101)

\[ P\lor\left(\lnot P\lor Q\right)\Leftrightarrow1 \]

(102)

\[ P\lor\left(\lnot P\land Q\right)\Leftrightarrow P\lor Q \]

(103)

\[ P\lor\left(\lnot P\rightarrow Q\right)\Leftrightarrow P\lor Q \]

(104)

\[ P\lor\left(\lnot P\leftarrow Q\right)\Leftrightarrow1 \]

(105)

\[ P\lor\left(\lnot P\leftrightarrow Q\right)\Leftrightarrow P\lor Q \]

(106)

\[ P\lor\left(\lnot P\downarrow Q\right)\Leftrightarrow P \]

(107)

\[ P\lor\left(\lnot P\uparrow Q\right)\Leftrightarrow \]

(108)

\[ P\lor\left(\lnot P\nrightarrow Q\right)\Leftrightarrow P\leftarrow Q \]

(109)

\[ P\lor\left(\lnot P\nleftarrow Q\right)\Leftrightarrow P \]

(110)

\[ P\lor\left(\lnot P\nleftrightarrow Q\right)\Leftrightarrow P\leftarrow Q \]

(111)

\[ P\land\left(\lnot P\lor Q\right)\Leftrightarrow P\land Q \]

(112)

\[ P\land\left(\lnot P\land Q\right)\Leftrightarrow0 \]

(113)

\[ P\land\left(\lnot P\rightarrow Q\right)\Leftrightarrow P \]

(114)

\[ P\land\left(\lnot P\leftarrow Q\right)\Leftrightarrow P\nrightarrow Q \]

(115)

\[ P\land\left(\lnot P\leftrightarrow Q\right)\Leftrightarrow P\nrightarrow Q \]

(116)

\[ P\land\left(\lnot P\downarrow Q\right)\Leftrightarrow P\nrightarrow Q \]

(117)

\[ P\land\left(\lnot P\uparrow Q\right)\Leftrightarrow P \]

(118)

\[ P\land\left(\lnot P\nrightarrow Q\right)\Leftrightarrow0 \]

(119)

\[ P\land\left(\lnot P\nleftarrow Q\right)\Leftrightarrow P\land Q \]

(120)

\[ P\land\left(\lnot P\nleftrightarrow Q\right)\Leftrightarrow P\land Q \]

(121)

\[ P\rightarrow\left(\lnot P\lor Q\right)\Leftrightarrow P\rightarrow Q \]

(122)

\[ P\rightarrow\left(\lnot P\land Q\right)\Leftrightarrow\lnot P \]

(123)

\[ P\rightarrow\left(\lnot P\rightarrow Q\right)\Leftrightarrow1 \]

(124)

\[ P\rightarrow\left(\lnot P\leftarrow Q\right)\Leftrightarrow P\uparrow Q \]

(125)

\[ P\rightarrow\left(\lnot P\leftrightarrow Q\right)\Leftrightarrow P\uparrow Q \]

(126)

\[ P\rightarrow\left(\lnot P\downarrow Q\right)\Leftrightarrow P\uparrow Q \]

(127)

\[ P\rightarrow\left(\lnot P\uparrow Q\right)\Leftrightarrow1 \]

(128)

\[ P\rightarrow\left(\lnot P\nrightarrow Q\right)\Leftrightarrow\lnot P \]

(129)

\[ P\rightarrow\left(\lnot P\nleftarrow Q\right)\Leftrightarrow P\rightarrow Q \]

(130)

\[ P\rightarrow\left(\lnot P\nleftrightarrow Q\right)\Leftrightarrow P\rightarrow Q \]

(131)

\[ P\leftarrow\left(\lnot P\lor Q\right)\Leftrightarrow P \]

(132)

\[ P\leftarrow\left(\lnot P\land Q\right)\Leftrightarrow P\leftarrow Q \]

(133)

\[ P\leftarrow\left(\lnot P\rightarrow Q\right)\Leftrightarrow P\leftarrow Q \]

(134)

\[ P\leftarrow\left(\lnot P\leftarrow Q\right)\Leftrightarrow P \]

(135)

\[ P\leftarrow\left(\lnot P\leftrightarrow Q\right)\Leftrightarrow P\leftarrow Q \]

(136)

\[ P\leftarrow\left(\lnot P\downarrow Q\right)\Leftrightarrow1 \]

(137)

\[ P\leftarrow\left(\lnot P\uparrow Q\right)\Leftrightarrow P\lor Q \]

(138)

\[ P\leftarrow\left(\lnot P\nrightarrow Q\right)\Leftrightarrow P\lor Q \]

(139)

\[ P\leftarrow\left(\lnot P\nleftarrow Q\right)\Leftrightarrow1 \]

(140)

\[ P\leftarrow\left(\lnot P\nleftrightarrow Q\right)\Leftrightarrow P\lor Q \]

(141)

\[ P\leftrightarrow\left(\lnot P\lor Q\right)\Leftrightarrow P\land Q \]

(142)

\[ P\leftrightarrow\left(\lnot P\land Q\right)\Leftrightarrow P\downarrow Q \]

(143)

\[ P\leftrightarrow\left(\lnot P\rightarrow Q\right)\Leftrightarrow P\leftarrow Q \]

(144)

\[ P\leftrightarrow\left(\lnot P\leftarrow Q\right)\Leftrightarrow P\nrightarrow Q \]

(145)

\[ P\leftrightarrow\left(\lnot P\leftrightarrow Q\right)\Leftrightarrow\lnot Q \]

(146)

\[ P\leftrightarrow\left(\lnot P\downarrow Q\right)\Leftrightarrow P\uparrow Q \]

(147)

\[ P\leftrightarrow\left(\lnot P\uparrow Q\right)\Leftrightarrow P\lor Q \]

(148)

\[ P\leftrightarrow\left(\lnot P\nrightarrow Q\right)\Leftrightarrow P\nleftarrow Q \]

(149)

\[ P\leftrightarrow\left(\lnot P\nleftarrow Q\right)\Leftrightarrow P\rightarrow Q \]

(150)

\[ P\leftrightarrow\left(\lnot P\nleftrightarrow Q\right)\Leftrightarrow Q \]

(151)

\[ P\downarrow\left(\lnot P\lor Q\right)\Leftrightarrow0 \]

(152)

\[ P\downarrow\left(\lnot P\land Q\right)\Leftrightarrow P\downarrow Q \]

(153)

\[ P\downarrow\left(\lnot P\rightarrow Q\right)\Leftrightarrow P\downarrow Q \]

(154)

\[ P\downarrow\left(\lnot P\leftarrow Q\right)\Leftrightarrow0 \]

(155)

\[ P\downarrow\left(\lnot P\leftrightarrow Q\right)\Leftrightarrow P\downarrow Q \]

(156)

\[ P\downarrow\left(\lnot P\downarrow Q\right)\Leftrightarrow\lnot P \]

(157)

\[ P\downarrow\left(\lnot P\uparrow Q\right)\Leftrightarrow\lnot P\land Q \]

(158)

\[ P\downarrow\left(\lnot P\nrightarrow Q\right)\Leftrightarrow P\nleftarrow Q \]

(159)

\[ P\downarrow\left(\lnot P\nleftarrow Q\right)\Leftrightarrow\lnot P \]

(160)

\[ P\downarrow\left(\lnot P\nleftrightarrow Q\right)\Leftrightarrow P\nleftarrow Q \]

(161)

\[ P\uparrow\left(\lnot P\lor Q\right)\Leftrightarrow P\uparrow Q \]

(162)

\[ P\uparrow\left(\lnot P\land Q\right)\Leftrightarrow1 \]

(163)

\[ P\uparrow\left(\lnot P\rightarrow Q\right)\Leftrightarrow\lnot P \]

(164)

\[ P\uparrow\left(\lnot P\leftarrow Q\right)\Leftrightarrow P\rightarrow Q \]

(165)

\[ P\uparrow\left(\lnot P\leftrightarrow Q\right)\Leftrightarrow P\rightarrow Q \]

(166)

\[ P\uparrow\left(\lnot P\downarrow Q\right)\Leftrightarrow P\rightarrow Q \]

(167)

\[ P\uparrow\left(\lnot P\uparrow Q\right)\Leftrightarrow\lnot P \]

(168)

\[ P\uparrow\left(\lnot P\nrightarrow Q\right)\Leftrightarrow1 \]

(169)

\[ P\uparrow\left(\lnot P\nleftarrow Q\right)\Leftrightarrow P\uparrow Q \]

(170)

\[ P\uparrow\left(\lnot P\nleftrightarrow Q\right)\Leftrightarrow P\uparrow Q \]

(171)

\[ P\nrightarrow\left(\lnot P\lor Q\right)\Leftrightarrow P\nrightarrow Q \]

(172)

\[ P\nrightarrow\left(\lnot P\land Q\right)\Leftrightarrow P \]

(173)

\[ P\nrightarrow\left(\lnot P\rightarrow Q\right)\Leftrightarrow0 \]

(174)

\[ P\nrightarrow\left(\lnot P\leftarrow Q\right)\Leftrightarrow P\land Q \]

(175)

\[ P\nrightarrow\left(\lnot P\leftrightarrow Q\right)\Leftrightarrow P\land Q \]

(176)

\[ P\nrightarrow\left(\lnot P\downarrow Q\right)\Leftrightarrow P\land Q \]

(177)

\[ P\nrightarrow\left(\lnot P\uparrow Q\right)\Leftrightarrow0 \]

(178)

\[ P\nrightarrow\left(\lnot P\nrightarrow Q\right)\Leftrightarrow P \]

(179)

\[ P\nrightarrow\left(\lnot P\nleftarrow Q\right)\Leftrightarrow P\nrightarrow Q \]

(180)

\[ P\nrightarrow\left(\lnot P\nleftrightarrow Q\right)\Leftrightarrow P\nrightarrow Q \]

(181)

\[ P\nleftarrow\left(\lnot P\lor Q\right)\Leftrightarrow\lnot P \]

(182)

\[ P\nleftarrow\left(\lnot P\land Q\right)\Leftrightarrow P\nleftarrow Q \]

(183)

\[ P\nleftarrow\left(\lnot P\rightarrow Q\right)\Leftrightarrow P\nleftarrow Q \]

(184)

\[ P\nleftarrow\left(\lnot P\leftarrow Q\right)\Leftrightarrow\lnot P \]

(185)

\[ P\nleftarrow\left(\lnot P\leftrightarrow Q\right)\Leftrightarrow P\nleftarrow Q \]

(186)

\[ P\nleftarrow\left(\lnot P\downarrow Q\right)\Leftrightarrow0 \]

(187)

\[ P\nleftarrow\left(\lnot P\uparrow Q\right)\Leftrightarrow P\downarrow Q \]

(188)

\[ P\nleftarrow\left(\lnot P\nrightarrow Q\right)\Leftrightarrow P\downarrow Q \]

(189)

\[ P\nleftarrow\left(\lnot P\nleftarrow Q\right)\Leftrightarrow0 \]

(190)

\[ P\nleftarrow\left(\lnot P\nleftrightarrow Q\right)\Leftrightarrow P\downarrow Q \]

(191)

\[ P\nleftrightarrow\left(\lnot P\lor Q\right)\Leftrightarrow P\uparrow Q \]

(192)

\[ P\nleftrightarrow\left(\lnot P\land Q\right)\Leftrightarrow P\lor Q \]

(193)

\[ P\nleftrightarrow\left(\lnot P\rightarrow Q\right)\Leftrightarrow P\nleftarrow Q \]

(194)

\[ P\nleftrightarrow\left(\lnot P\leftarrow Q\right)\Leftrightarrow P\rightarrow Q \]

(195)

\[ P\nleftrightarrow\left(\lnot P\leftrightarrow Q\right)\Leftrightarrow Q \]

(196)

\[ P\nleftrightarrow\left(\lnot P\downarrow Q\right)\Leftrightarrow P\land Q \]

(197)

\[ P\nleftrightarrow\left(\lnot P\uparrow Q\right)\Leftrightarrow P\downarrow Q \]

(198)

\[ P\nleftrightarrow\left(\lnot P\nrightarrow Q\right)\Leftrightarrow P\leftarrow Q \]

(199)

\[ P\nleftrightarrow\left(\lnot P\nleftarrow Q\right)\Leftrightarrow P\nrightarrow Q \]

(200)

\[ P\nleftrightarrow\left(\lnot P\nleftrightarrow Q\right)\Leftrightarrow\lnot Q \]

一覧

\[ \begin{array}{|c|c|c|c|} \hline L_{1} & L_{2} & PL_{1}\left(PL_{2}Q\right) & PL_{1}\left(\lnot PL_{2}Q\right)\\ \hline \lor & \lor & P\lor Q & 1\\ \hline \lor & \land & P & P\lor Q\\ \hline \lor & \rightarrow & 1 & P\lor Q\\ \hline \lor & \leftarrow & P\leftarrow Q & 1\\ \hline \lor & \leftrightarrow & P\leftarrow Q & P\lor Q\\ \hline \lor & \downarrow & P\leftarrow Q & P\\ \hline \lor & \uparrow & 1 & P\leftarrow Q\\ \hline \lor & \nrightarrow & P & P\leftarrow Q\\ \hline \lor & \nleftarrow & P\lor Q & P\\ \hline \lor & \nleftrightarrow & P\lor Q & P\leftarrow Q\\ \hline \land & \lor & P & P\land Q\\ \hline \land & \land & P\land Q & 0\\ \hline \land & \rightarrow & P\land Q & P\\ \hline \land & \leftarrow & P & P\nrightarrow Q\\ \hline \land & \leftrightarrow & P\land Q & P\nrightarrow Q\\ \hline \land & \downarrow & 0 & P\nrightarrow Q\\ \hline \land & \uparrow & P\nrightarrow Q & P\\ \hline \land & \nrightarrow & P\nrightarrow Q & 0\\ \hline \land & \nleftarrow & 0 & P\land Q\\ \hline \land & \nleftrightarrow & P\nrightarrow Q & P\land Q\\ \hline \rightarrow & \lor & 1 & P\rightarrow Q\\ \hline \rightarrow & \land & P\rightarrow Q & \lnot P\\ \hline \rightarrow & \rightarrow & P\rightarrow Q & 1\\ \hline \rightarrow & \leftarrow & 1 & P\uparrow Q\\ \hline \rightarrow & \leftrightarrow & P\rightarrow Q & P\uparrow Q\\ \hline \rightarrow & \downarrow & \lnot P & P\uparrow Q\\ \hline \rightarrow & \uparrow & P\uparrow Q & 1\\ \hline \rightarrow & \nrightarrow & P\uparrow Q & \lnot P\\ \hline \rightarrow & \nleftarrow & \lnot P & P\rightarrow Q\\ \hline \rightarrow & \nleftrightarrow & P\uparrow Q & P\rightarrow Q\\ \hline \leftarrow & \lor & P\leftarrow Q & P\\ \hline \leftarrow & \land & 1 & P\leftarrow Q\\ \hline \leftarrow & \rightarrow & P & P\leftarrow Q\\ \hline \leftarrow & \leftarrow & P\lor Q & P\\ \hline \leftarrow & \leftrightarrow & P\lor Q & P\leftarrow Q\\ \hline \leftarrow & \downarrow & P\lor Q & 1\\ \hline \leftarrow & \uparrow & P & P\lor Q\\ \hline \leftarrow & \nrightarrow & 1 & P\lor Q\\ \hline \leftarrow & \nleftarrow & P\leftarrow Q & 1\\ \hline \leftarrow & \nleftrightarrow & P\leftarrow Q & P\lor Q\\ \hline \leftrightarrow & \lor & P\leftarrow Q & P\land Q\\ \hline \leftrightarrow & \land & P\rightarrow Q & P\downarrow Q\\ \hline \leftrightarrow & \rightarrow & P\land Q & P\leftarrow Q\\ \hline \leftrightarrow & \leftarrow & P\lor Q & P\nrightarrow Q\\ \hline \leftrightarrow & \leftrightarrow & Q & \lnot Q\\ \hline \leftrightarrow & \downarrow & P\nleftarrow Q & P\uparrow Q\\ \hline \leftrightarrow & \uparrow & P\nrightarrow Q & P\lor Q\\ \hline \leftrightarrow & \nrightarrow & P\uparrow Q & P\nleftarrow Q\\ \hline \leftrightarrow & \nleftarrow & P\downarrow Q & P\rightarrow Q\\ \hline \leftrightarrow & \nleftrightarrow & \lnot Q & Q\\ \hline \downarrow & \lor & P\downarrow Q & 0\\ \hline \downarrow & \land & \lnot P & P\downarrow Q\\ \hline \downarrow & \rightarrow & 0 & P\downarrow Q\\ \hline \downarrow & \leftarrow & P\nleftarrow Q & 0\\ \hline \downarrow & \leftrightarrow & P\nleftarrow Q & P\downarrow Q\\ \hline \downarrow & \downarrow & P\nleftarrow Q & \lnot P\\ \hline \downarrow & \uparrow & 0 & \lnot P\land Q\\ \hline \downarrow & \nrightarrow & \lnot P & P\nleftarrow Q\\ \hline \downarrow & \nleftarrow & P\downarrow Q & \lnot P\\ \hline \downarrow & \nleftrightarrow & P\downarrow Q & P\nleftarrow Q\\ \hline \uparrow & \lor & \lnot P & P\uparrow Q\\ \hline \uparrow & \land & P\uparrow Q & 1\\ \hline \uparrow & \rightarrow & P\uparrow Q & \lnot P\\ \hline \uparrow & \leftarrow & \lnot P & P\rightarrow Q\\ \hline \uparrow & \leftrightarrow & P\uparrow Q & P\rightarrow Q\\ \hline \uparrow & \downarrow & 1 & P\rightarrow Q\\ \hline \uparrow & \uparrow & P\rightarrow Q & \lnot P\\ \hline \uparrow & \nrightarrow & P\rightarrow Q & 1\\ \hline \uparrow & \nleftarrow & 1 & P\uparrow Q\\ \hline \uparrow & \nleftrightarrow & P\rightarrow Q & P\uparrow Q\\ \hline \nrightarrow & \lor & 0 & P\nrightarrow Q\\ \hline \nrightarrow & \land & P\nrightarrow Q & P\\ \hline \nrightarrow & \rightarrow & P\nrightarrow Q & 0\\ \hline \nrightarrow & \leftarrow & 0 & P\land Q\\ \hline \nrightarrow & \leftrightarrow & P\nrightarrow Q & P\land Q\\ \hline \nrightarrow & \downarrow & P & P\land Q\\ \hline \nrightarrow & \uparrow & P\land Q & 0\\ \hline \nrightarrow & \nrightarrow & P\land Q & P\\ \hline \nrightarrow & \nleftarrow & P & P\nrightarrow Q\\ \hline \nrightarrow & \nleftrightarrow & P\land Q & P\nrightarrow Q\\ \hline \nleftarrow & \lor & P\nleftarrow Q & \lnot P\\ \hline \nleftarrow & \land & 0 & P\nleftarrow Q\\ \hline \nleftarrow & \rightarrow & \lnot P & P\nleftarrow Q\\ \hline \nleftarrow & \leftarrow & P\downarrow Q & \lnot P\\ \hline \nleftarrow & \leftrightarrow & P\downarrow Q & P\nleftarrow Q\\ \hline \nleftarrow & \downarrow & P\downarrow Q & 0\\ \hline \nleftarrow & \uparrow & \lnot P & P\downarrow Q\\ \hline \nleftarrow & \nrightarrow & 0 & P\downarrow Q\\ \hline \nleftarrow & \nleftarrow & P\nleftarrow Q & 0\\ \hline \nleftarrow & \nleftrightarrow & P\nleftarrow Q & P\downarrow Q\\ \hline \nleftrightarrow & \lor & P\nleftarrow Q & P\uparrow Q\\ \hline \nleftrightarrow & \land & P\nrightarrow Q & P\lor Q\\ \hline \nleftrightarrow & \rightarrow & P\uparrow Q & P\nleftarrow Q\\ \hline \nleftrightarrow & \leftarrow & P\downarrow Q & P\rightarrow Q\\ \hline \nleftrightarrow & \leftrightarrow & \lnot Q & Q\\ \hline \nleftrightarrow & \downarrow & P\leftarrow Q & P\land Q\\ \hline \nleftrightarrow & \uparrow & P\rightarrow Q & P\downarrow Q\\ \hline \nleftrightarrow & \nrightarrow & P\land Q & P\leftarrow Q\\ \hline \nleftrightarrow & \nleftarrow & P\lor Q & P\nrightarrow Q\\ \hline \nleftrightarrow & \nleftrightarrow & Q & \lnot Q \\\hline \end{array} \]

(1)

\begin{align*} P\lor\left(P\lor Q\right) & \Leftrightarrow\left(P\lor P\right)\lor Q\\ & \Leftrightarrow P\lor Q \end{align*}

(2)

吸収律より、

\[ P\lor\left(P\land Q\right)\Leftrightarrow P \]

が成り立つ。

(3)

\begin{align*} P\lor\left(P\rightarrow Q\right) & \Leftrightarrow P\lor\left(\lnot P\lor Q\right)\\ & \Leftrightarrow\left(P\lor\lnot P\right)\lor Q\\ & \Leftrightarrow1\lor Q\\ & \Leftrightarrow1 \end{align*}

(4)

\begin{align*} P\lor\left(P\leftarrow Q\right) & \Leftrightarrow P\lor\left(P\lor\lnot Q\right)\\ & \Leftrightarrow\left(P\lor P\right)\lor\lnot Q\\ & \Leftrightarrow P\lor\lnot Q\\ & \Leftrightarrow P\leftarrow Q \end{align*}

(5)

\begin{align*} P\lor\left(P\leftrightarrow Q\right) & \Leftrightarrow P\lor\left\{ \left(\lnot P\lor Q\right)\land\left(P\lor\lnot Q\right)\right\} \\ & \Leftrightarrow P\lor\left\{ \left(\lnot P\lor Q\right)\land\left(P\lor\lnot Q\right)\right\} \\ & \Leftrightarrow P\lor\left(P\lor\lnot Q\right)\\ & \Leftrightarrow\left(P\lor P\right)\lor\lnot Q\\ & \Leftrightarrow P\lor\lnot Q\\ & \Leftrightarrow P\leftarrow Q \end{align*}

(6)

\begin{align*} P\lor\left(P\downarrow Q\right) & \Leftrightarrow P\lor\left(\lnot P\land\lnot Q\right)\\ & \Leftrightarrow P\lor\lnot Q\\ & \Leftrightarrow P\leftarrow Q \end{align*}

(7)

\begin{align*} P\lor\left(P\uparrow Q\right) & \Leftrightarrow P\lor\left(\lnot P\lor\lnot Q\right)\\ & \Leftrightarrow\left(P\lor\lnot P\right)\lor\lnot Q\\ & \Leftrightarrow1 \end{align*}

(8)

\begin{align*} P\lor\left(P\nrightarrow Q\right) & \Leftrightarrow P\lor\left(P\land\lnot Q\right)\\ & \Leftrightarrow P \end{align*}

(9)

\begin{align*} P\lor\left(P\nleftarrow Q\right) & \Leftrightarrow P\lor\left(\lnot P\land Q\right)\\ & \Leftrightarrow P\lor Q \end{align*}

(10)

\begin{align*} P\lor\left(P\nleftrightarrow Q\right) & \Leftrightarrow P\lor\left\{ \left(P\land\lnot Q\right)\lor\left(\lnot P\land Q\right)\right\} \\ & \Leftrightarrow P\lor\left(P\land\lnot Q\right)\lor\left(\lnot P\land Q\right)\\ & \Leftrightarrow P\lor\left(\lnot P\land Q\right)\\ & \Leftrightarrow P\lor Q \end{align*}

(11)

吸収律より、

\[ P\land\left(P\lor Q\right)\Leftrightarrow P \]

が成り立つ。

(12)

\begin{align*} P\land\left(P\land Q\right) & \Leftrightarrow\left(P\land P\right)\land Q\\ & \Leftrightarrow P\land Q \end{align*}

(13)

\begin{align*} P\land\left(P\rightarrow Q\right) & \Leftrightarrow P\land\left(\lnot P\lor Q\right)\\ & \Leftrightarrow P\land Q \end{align*}

(14)

\begin{align*} P\land\left(P\leftarrow Q\right) & \Leftrightarrow P\land\left(P\lor\lnot Q\right)\\ & \Leftrightarrow P \end{align*}

(15)

\begin{align*} P\land\left(P\leftrightarrow Q\right) & \Leftrightarrow P\land\left\{ \left(\lnot P\lor Q\right)\land\left(P\lor\lnot Q\right)\right\} \\ & \Leftrightarrow P\land\left(\lnot P\lor Q\right)\land\left(P\lor\lnot Q\right)\\ & \Leftrightarrow P\land\left(\lnot P\lor Q\right)\\ & \Leftrightarrow P\land Q \end{align*}

(16)

\begin{align*} P\land\left(P\downarrow Q\right) & \Leftrightarrow P\land\left(\lnot P\land\lnot Q\right)\\ & \Leftrightarrow\left(P\land\lnot P\right)\land\lnot Q\\ & \Leftrightarrow0 \end{align*}

(17)

\begin{align*} P\land\left(P\uparrow Q\right) & \Leftrightarrow P\land\left(\lnot P\lor\lnot Q\right)\\ & \Leftrightarrow P\land\lnot Q\\ & \Leftrightarrow P\nrightarrow Q \end{align*}

(18)

\begin{align*} P\land\left(P\nrightarrow Q\right) & \Leftrightarrow P\land\left(P\land\lnot Q\right)\\ & \Leftrightarrow\left(P\land P\right)\land\lnot Q\\ & \Leftrightarrow P\nrightarrow Q \end{align*}

(19)

\begin{align*} P\land\left(P\nleftarrow Q\right) & \Leftrightarrow P\land\left(\lnot P\land Q\right)\\ & \Leftrightarrow\left(P\land\lnot P\right)\land Q\\ & \Leftrightarrow0 \end{align*}

(20)

\begin{align*} P\land\left(P\nleftrightarrow Q\right) & \Leftrightarrow P\land\left\{ \left(P\land\lnot Q\right)\lor\left(\lnot P\land Q\right)\right\} \\ & \Leftrightarrow P\land\left(P\land\lnot Q\right)\\ & \Leftrightarrow\left(P\land P\right)\land\lnot Q\\ & \Leftrightarrow P\land\lnot Q\\ & \Leftrightarrow P\nrightarrow Q \end{align*}

(21)

\begin{align*} P\rightarrow\left(P\lor Q\right) & \Leftrightarrow\lnot P\lor\left(P\lor Q\right)\\ & \Leftrightarrow\left(\lnot P\lor P\right)\lor Q\\ & \Leftrightarrow1 \end{align*}

(22)

\begin{align*} P\rightarrow\left(P\land Q\right) & \Leftrightarrow\lnot P\lor\left(P\land Q\right)\\ & \Leftrightarrow\lnot P\lor Q\\ & \Leftrightarrow P\rightarrow Q \end{align*}

(23)

\begin{align*} P\rightarrow\left(P\rightarrow Q\right) & \Leftrightarrow\lnot P\lor\left(\lnot P\lor Q\right)\\ & \Leftrightarrow\left(\lnot P\lor\lnot P\right)\lor Q\\ & \Leftrightarrow P\rightarrow Q \end{align*}

(24)

\begin{align*} P\rightarrow\left(P\leftarrow Q\right) & \Leftrightarrow\lnot P\lor\left(P\lor\lnot Q\right)\\ & \Leftrightarrow\left(\lnot P\lor P\right)\lor\lnot Q\\ & \Leftrightarrow1 \end{align*}

(25)

\begin{align*} P\rightarrow\left(P\leftrightarrow Q\right) & \Leftrightarrow\lnot P\lor\left\{ \left(\lnot P\lor Q\right)\land\left(P\lor\lnot Q\right)\right\} \\ & \Leftrightarrow\lnot P\lor\left(\lnot P\lor Q\right)\\ & \Leftrightarrow\left(\lnot P\lor\lnot P\right)\lor Q\\ & \Leftrightarrow\lnot P\lor Q\\ & \Leftrightarrow P\rightarrow Q \end{align*}

(26)

\begin{align*} P\rightarrow\left(P\downarrow Q\right) & \Leftrightarrow\lnot P\lor\left(\lnot P\land\lnot Q\right)\\ & \Leftrightarrow\lnot P \end{align*}

(27)

\begin{align*} P\rightarrow\left(P\uparrow Q\right) & \Leftrightarrow\lnot P\lor\left(\lnot P\lor\lnot Q\right)\\ & \Leftrightarrow\left(\lnot P\lor\lnot P\right)\lor\lnot Q\\ & \Leftrightarrow P\uparrow Q \end{align*}

(28)

\begin{align*} P\rightarrow\left(P\nrightarrow Q\right) & \Leftrightarrow\lnot P\lor\left(P\land\lnot Q\right)\\ & \Leftrightarrow\lnot P\lor\lnot Q\\ & \Leftrightarrow P\uparrow Q \end{align*}

(29)

\begin{align*} P\rightarrow\left(P\nleftarrow Q\right) & \Leftrightarrow\lnot P\lor\left(\lnot P\land Q\right)\\ & \Leftrightarrow\lnot P \end{align*}

(30)

\begin{align*} P\rightarrow\left(P\nleftrightarrow Q\right) & \Leftrightarrow\lnot P\lor\left\{ \left(P\land\lnot Q\right)\lor\left(\lnot P\land Q\right)\right\} \\ & \Leftrightarrow\lnot P\lor\left(P\land\lnot Q\right)\lor\left(\lnot P\land Q\right)\\ & \Leftrightarrow\lnot P\lor\left(P\land\lnot Q\right)\\ & \Leftrightarrow\lnot P\lor\lnot Q\\ & \Leftrightarrow P\uparrow Q \end{align*}

(31)

\begin{align*} P\leftarrow\left(P\lor Q\right) & \Leftrightarrow P\lor\lnot\left(P\lor Q\right)\\ & \Leftrightarrow P\lor\left(\lnot P\land\lnot Q\right)\\ & \Leftrightarrow P\lor\lnot Q\\ & \Leftrightarrow P\leftarrow Q \end{align*}

(32)

\begin{align*} P\leftarrow\left(P\land Q\right) & \Leftrightarrow P\lor\lnot\left(P\land Q\right)\\ & \Leftrightarrow P\lor\left(\lnot P\lor\lnot Q\right)\\ & \Leftrightarrow\left(P\lor\lnot P\right)\lor\lnot Q\\ & \Leftrightarrow1 \end{align*}

(33)

\begin{align*} P\leftarrow\left(P\rightarrow Q\right) & \Leftrightarrow P\lor\lnot\left(\lnot P\lor Q\right)\\ & \Leftrightarrow P\lor\left(P\land\lnot Q\right)\\ & \Leftrightarrow P \end{align*}

(34)

\begin{align*} P\leftarrow\left(P\leftarrow Q\right) & \Leftrightarrow P\lor\left(P\nleftarrow Q\right)\\ & \Leftrightarrow P\lor\left(\lnot P\land Q\right)\\ & \Leftrightarrow P\lor Q \end{align*}

(35)

\begin{align*} P\leftarrow\left(P\leftrightarrow Q\right) & \Leftrightarrow P\lor\lnot\left(P\leftrightarrow Q\right)\\ & \Leftrightarrow P\lor\left(P\nleftrightarrow Q\right)\\ & \Leftrightarrow P\lor\left\{ \left(P\land\lnot Q\right)\lor\left(\lnot P\land Q\right)\right\} \\ & \Leftrightarrow P\lor\left(P\land\lnot Q\right)\lor\left(\lnot P\land Q\right)\\ & \Leftrightarrow P\lor\left(\lnot P\land Q\right)\\ & \Leftrightarrow P\lor Q \end{align*}

(36)

\begin{align*} P\leftarrow\left(P\downarrow Q\right) & \Leftrightarrow P\lor\lnot\left(P\downarrow Q\right)\\ & \Leftrightarrow P\lor\left(P\lor Q\right)\\ & \Leftrightarrow\left(P\lor P\right)\lor Q\\ & \Leftrightarrow P\lor Q \end{align*}

(37)

\begin{align*} P\leftarrow\left(P\uparrow Q\right) & \Leftrightarrow P\lor\lnot\left(P\uparrow Q\right)\\ & \Leftrightarrow P\lor\left(P\land Q\right)\\ & \Leftrightarrow P \end{align*}

(38)

\begin{align*} P\leftarrow\left(P\nrightarrow Q\right) & \Leftrightarrow P\lor\lnot\left(P\nrightarrow Q\right)\\ & \Leftrightarrow P\lor\left(P\rightarrow Q\right)\\ & \Leftrightarrow P\lor\left(\lnot P\lor Q\right)\\ & \Leftrightarrow\left(P\lor\lnot P\right)\lor Q\\ & \Leftrightarrow1 \end{align*}

(39)

\begin{align*} P\leftarrow\left(P\nleftarrow Q\right) & \Leftrightarrow P\lor\lnot\left(P\nleftarrow Q\right)\\ & \Leftrightarrow P\lor\left(P\leftarrow Q\right)\\ & \Leftrightarrow P\lor\left(P\lor\lnot Q\right)\\ & \Leftrightarrow\left(P\lor P\right)\lor\lnot Q\\ & \Leftrightarrow P\leftarrow Q \end{align*}

(40)

\begin{align*} P\leftarrow\left(P\nleftrightarrow Q\right) & \Leftrightarrow P\lor\left(P\leftrightarrow Q\right)\\ & \Leftrightarrow P\lor\left\{ \left(\lnot P\lor Q\right)\land\left(P\lor\lnot Q\right)\right\} \\ & \Leftrightarrow P\lor\left(P\lor\lnot Q\right)\\ & \Leftrightarrow\left(P\lor P\right)\lor\lnot Q\\ & \Leftrightarrow P\leftarrow Q \end{align*}

(41)

\begin{align*} P\leftrightarrow\left(P\lor Q\right) & \Leftrightarrow\left\{ \lnot P\lor\left(P\lor Q\right)\right\} \land\left\{ P\lor\lnot\left(P\lor Q\right)\right\} \\ & \Leftrightarrow\left\{ \left(\lnot P\lor P\right)\lor Q\right\} \land\left\{ P\lor\left(\lnot P\land\lnot Q\right)\right\} \\ & \Leftrightarrow P\lor\left(\lnot P\land\lnot Q\right)\\ & \Leftrightarrow P\lor\lnot Q\\ & \Leftrightarrow P\leftarrow Q \end{align*}

(42)

\begin{align*} P\leftrightarrow\left(P\land Q\right) & \Leftrightarrow\left\{ \lnot P\lor\left(P\land Q\right)\right\} \land\left\{ P\lor\lnot\left(P\land Q\right)\right\} \\ & \Leftrightarrow\left(\lnot P\lor Q\right)\land\left\{ P\lor\left(\lnot P\lor\lnot Q\right)\right\} \\ & \Leftrightarrow\left(\lnot P\lor Q\right)\land\left\{ \left(P\lor\lnot P\right)\lor\lnot Q\right\} \\ & \Leftrightarrow\lnot P\lor Q\\ & \Leftrightarrow P\rightarrow Q \end{align*}

(43)

\begin{align*} P\leftrightarrow\left(P\rightarrow Q\right) & \Leftrightarrow\left\{ \lnot P\lor\left(P\rightarrow Q\right)\right\} \land\left\{ P\lor\lnot\left(P\rightarrow Q\right)\right\} \\ & \Leftrightarrow\left\{ \lnot P\lor\left(\lnot P\lor Q\right)\right\} \land\left\{ P\lor\left(P\nrightarrow Q\right)\right\} \\ & \Leftrightarrow\left\{ \left(\lnot P\lor\lnot P\right)\lor Q\right\} \land\left\{ P\lor\left(P\land\lnot Q\right)\right\} \\ & \Leftrightarrow\left(\lnot P\lor Q\right)\land\left\{ P\lor\left(P\land\lnot Q\right)\right\} \\ & \Leftrightarrow\left(\lnot P\lor Q\right)\land P\\ & \Leftrightarrow P\land Q \end{align*}

(44)

\begin{align*} P\leftrightarrow\left(P\leftarrow Q\right) & \Leftrightarrow\left\{ \lnot P\lor\left(P\leftarrow Q\right)\right\} \land\left\{ P\lor\lnot\left(P\leftarrow Q\right)\right\} \\ & \Leftrightarrow\left\{ \lnot P\lor\left(P\lor\lnot Q\right)\right\} \land\left\{ P\lor\left(P\nleftarrow Q\right)\right\} \\ & \Leftrightarrow\left\{ \left(\lnot P\lor P\right)\lor\lnot Q\right\} \land\left\{ P\lor\left(\lnot P\land Q\right)\right\} \\ & \Leftrightarrow P\lor\left(\lnot P\land Q\right)\\ & \Leftrightarrow P\lor Q \end{align*}

(45)

\begin{align*} P\leftrightarrow\left(P\leftrightarrow Q\right) & \Leftrightarrow\left(P\leftrightarrow P\right)\leftrightarrow Q\\ & \Leftrightarrow1\leftrightarrow Q\\ & \Leftrightarrow Q \end{align*}

(46)

\begin{align*} P\leftrightarrow\left(P\downarrow Q\right) & \Leftrightarrow\left\{ \lnot P\lor\left(P\downarrow Q\right)\right\} \land\left\{ P\lor\lnot\left(P\downarrow Q\right)\right\} \\ & \Leftrightarrow\left\{ \lnot P\lor\left(\lnot P\land\lnot Q\right)\right\} \land\left\{ P\lor\left(P\lor Q\right)\right\} \\ & \Leftrightarrow\lnot P\land\left\{ \left(P\lor P\right)\lor Q\right\} \\ & \Leftrightarrow\lnot P\land\left(P\lor Q\right)\\ & \Leftrightarrow\lnot P\land Q\\ & \Leftrightarrow P\nleftarrow Q \end{align*}

(47)

\begin{align*} P\leftrightarrow\left(P\uparrow Q\right) & \Leftrightarrow\left\{ \lnot P\lor\left(P\uparrow Q\right)\right\} \land\left\{ P\lor\lnot\left(P\uparrow Q\right)\right\} \\ & \Leftrightarrow\left\{ \lnot P\lor\left(\lnot P\lor\lnot Q\right)\right\} \land\left\{ P\lor\left(P\land Q\right)\right\} \\ & \Leftrightarrow\left\{ \left(\lnot P\lor\lnot P\right)\lor\lnot Q\right\} \land P\\ & \Leftrightarrow\left(\lnot P\lor\lnot Q\right)\land P\\ & \Leftrightarrow\lnot Q\land P\\ & \Leftrightarrow P\nrightarrow Q \end{align*}

(48)

\begin{align*} P\leftrightarrow\left(P\nrightarrow Q\right) & \Leftrightarrow\left\{ \lnot P\lor\left(P\nrightarrow Q\right)\right\} \land\left\{ P\lor\lnot\left(P\nrightarrow Q\right)\right\} \\ & \Leftrightarrow\left\{ \lnot P\lor\left(P\land\lnot Q\right)\right\} \land\left\{ P\lor\left(P\rightarrow Q\right)\right\} \\ & \Leftrightarrow\left(\lnot P\lor\lnot Q\right)\land\left\{ P\lor\left(\lnot P\lor Q\right)\right\} \\ & \Leftrightarrow\left(\lnot P\lor\lnot Q\right)\land\left\{ \left(P\lor\lnot P\right)\lor Q\right\} \\ & \Leftrightarrow\left(\lnot P\lor\lnot Q\right)\\ & \Leftrightarrow P\uparrow Q \end{align*}

(49)

\begin{align*} P\leftrightarrow\left(P\nleftarrow Q\right) & \Leftrightarrow\left\{ \lnot P\lor\left(P\nleftarrow Q\right)\right\} \land\left\{ P\lor\left(P\leftarrow Q\right)\right\} \\ & \Leftrightarrow\left\{ \lnot P\lor\left(\lnot P\land Q\right)\right\} \land\left\{ P\lor\left(P\lor\lnot Q\right)\right\} \\ & \Leftrightarrow\lnot P\land\left\{ \left(P\lor P\right)\lor\lnot Q\right\} \\ & \Leftrightarrow\lnot P\land\left(P\lor\lnot Q\right)\\ & \Leftrightarrow\lnot P\land\lnot Q\\ & \Leftrightarrow P\downarrow Q \end{align*}

(50)

\begin{align*} P\leftrightarrow\left(P\nleftrightarrow Q\right) & \Leftrightarrow\left(P\leftrightarrow P\right)\nleftrightarrow Q\\ & \Leftrightarrow1\nleftrightarrow Q\\ & \Leftrightarrow\lnot Q \end{align*}

(51)

\begin{align*} P\downarrow\left(P\lor Q\right) & \Leftrightarrow\lnot P\land\lnot\left(P\lor Q\right)\\ & \Leftrightarrow\lnot P\land\left(\lnot P\land\lnot Q\right)\\ & \Leftrightarrow\left(\lnot P\land\lnot P\right)\land\lnot Q\\ & \Leftrightarrow P\downarrow Q \end{align*}

(52)

\begin{align*} P\downarrow\left(P\land Q\right) & \Leftrightarrow\lnot P\land\lnot\left(P\land Q\right)\\ & \Leftrightarrow\lnot P\land\left(\lnot P\lor\lnot Q\right)\\ & \Leftrightarrow\lnot P \end{align*}

(53)

\begin{align*} P\downarrow\left(P\rightarrow Q\right) & \Leftrightarrow\lnot P\land\left(P\nrightarrow Q\right)\\ & \Leftrightarrow\lnot P\land\left(P\land\lnot Q\right)\\ & \Leftrightarrow\left(\lnot P\land P\right)\land\lnot Q\\ & \Leftrightarrow0 \end{align*}

(54)

\begin{align*} P\downarrow\left(P\leftarrow Q\right) & \Leftrightarrow\lnot P\land\left(P\nleftarrow Q\right)\\ & \Leftrightarrow\lnot P\land\left(\lnot P\land Q\right)\\ & \Leftrightarrow\left(\lnot P\land\lnot P\right)\land Q\\ & \Leftrightarrow P\nleftarrow Q \end{align*}

(55)

\begin{align*} P\downarrow\left(P\leftrightarrow Q\right) & \Leftrightarrow\lnot P\land\lnot\left(P\leftrightarrow Q\right)\\ & \Leftrightarrow\lnot P\land\left(P\nleftrightarrow Q\right)\\ & \Leftrightarrow\lnot P\land\left\{ \left(P\land\lnot Q\right)\lor\left(\lnot P\land Q\right)\right\} \\ & \Leftrightarrow\lnot P\land\left(\lnot P\land Q\right)\\ & \Leftrightarrow\left(\lnot P\land\lnot P\right)\land Q\\ & \Leftrightarrow P\nleftarrow Q \end{align*}

(56)

\begin{align*} P\downarrow\left(P\downarrow Q\right) & \Leftrightarrow\lnot P\land\lnot\left(P\downarrow Q\right)\\ & \Leftrightarrow\lnot P\land\left(P\lor Q\right)\\ & \Leftrightarrow\lnot P\land Q\\ & \Leftrightarrow P\nleftarrow Q \end{align*}

(57)

\begin{align*} P\downarrow\left(P\uparrow Q\right) & \Leftrightarrow\lnot P\land\lnot\left(P\uparrow Q\right)\\ & \Leftrightarrow\lnot P\land\left(P\land Q\right)\\ & \Leftrightarrow\left(\lnot P\land P\right)\land Q\\ & \Leftrightarrow0 \end{align*}

(58)

\begin{align*} P\downarrow\left(P\nrightarrow Q\right) & \Leftrightarrow\lnot P\land\lnot\left(P\nrightarrow Q\right)\\ & \Leftrightarrow\lnot P\land\left(P\rightarrow Q\right)\\ & \Leftrightarrow\lnot P\land\left(\lnot P\lor Q\right)\\ & \Leftrightarrow\lnot P \end{align*}

(59)

\begin{align*} P\downarrow\left(P\nleftarrow Q\right) & \Leftrightarrow\lnot P\land\lnot\left(P\nleftarrow Q\right)\\ & \Leftrightarrow\lnot P\land\left(P\leftarrow Q\right)\\ & \Leftrightarrow\lnot P\land\left(P\lor\lnot Q\right)\\ & \Leftrightarrow\lnot P\land\lnot Q\\ & \Leftrightarrow P\downarrow Q \end{align*}

(60)

\begin{align*} P\downarrow\left(P\nleftrightarrow Q\right) & \Leftrightarrow\lnot P\land\lnot\left(P\nleftrightarrow Q\right)\\ & \Leftrightarrow\lnot P\land\left(P\leftrightarrow Q\right)\\ & \Leftrightarrow\lnot P\land\left\{ \left(P\lor\lnot Q\right)\land\left(\lnot P\lor Q\right)\right\} \\ & \Leftrightarrow\lnot P\land\left(P\lor\lnot Q\right)\land\left(\lnot P\lor Q\right)\\ & \Leftrightarrow\lnot P\land\left(P\lor\lnot Q\right)\\ & \Leftrightarrow\lnot P\land\lnot Q\\ & \Leftrightarrow P\downarrow Q \end{align*}

(61)

\begin{align*} P\uparrow\left(P\lor Q\right) & \Leftrightarrow\lnot P\lor\lnot\left(P\lor Q\right)\\ & \Leftrightarrow\lnot P\lor\left(\lnot P\land\lnot Q\right)\\ & \Leftrightarrow\lnot P \end{align*}

(62)

\begin{align*} P\uparrow\left(P\land Q\right) & \Leftrightarrow\lnot P\lor\lnot\left(P\land Q\right)\\ & \Leftrightarrow\lnot P\lor\left(\lnot P\lor\lnot Q\right)\\ & \Leftrightarrow\left(\lnot P\lor\lnot P\right)\lor\lnot Q\\ & \Leftrightarrow P\uparrow Q \end{align*}

(63)

\begin{align*} P\uparrow\left(P\rightarrow Q\right) & \Leftrightarrow\lnot P\lor\lnot\left(P\rightarrow Q\right)\\ & \Leftrightarrow\lnot P\lor\left(P\nrightarrow Q\right)\\ & \Leftrightarrow\lnot P\lor\left(P\land\lnot Q\right)\\ & \Leftrightarrow\lnot P\lor\lnot Q\\ & \Leftrightarrow P\uparrow Q \end{align*}

(64)

\begin{align*} P\uparrow\left(P\leftarrow Q\right) & \Leftrightarrow\lnot P\lor\lnot\left(P\leftarrow Q\right)\\ & \Leftrightarrow\lnot P\lor\left(P\nleftarrow Q\right)\\ & \Leftrightarrow\lnot P\lor\left(\lnot P\land Q\right)\\ & \Leftrightarrow\lnot P \end{align*}

(65)

\begin{align*} P\uparrow\left(P\leftrightarrow Q\right) & \Leftrightarrow\lnot P\lor\lnot\left(P\leftrightarrow Q\right)\\ & \Leftrightarrow\lnot P\lor\left(P\nleftrightarrow Q\right)\\ & \Leftrightarrow\lnot P\lor\left\{ \left(P\land\lnot Q\right)\lor\left(\lnot P\land Q\right)\right\} \\ & \Leftrightarrow\lnot P\lor\left(P\land\lnot Q\right)\lor\left(\lnot P\land Q\right)\\ & \Leftrightarrow\lnot P\lor\left(P\land\lnot Q\right)\\ & \Leftrightarrow\lnot P\lor\lnot Q\\ & \Leftrightarrow P\uparrow Q \end{align*}

(66)

\begin{align*} P\uparrow\left(P\downarrow Q\right) & \Leftrightarrow\lnot P\lor\lnot\left(P\downarrow Q\right)\\ & \Leftrightarrow\lnot P\lor\left(P\lor Q\right)\\ & \Leftrightarrow\left(\lnot P\lor P\right)\lor Q\\ & \Leftrightarrow1 \end{align*}

(67)

\begin{align*} P\uparrow\left(P\uparrow Q\right) & \Leftrightarrow\lnot P\lor\left(P\land Q\right)\\ & \Leftrightarrow\lnot P\lor Q\\ & \Leftrightarrow P\rightarrow Q \end{align*}

(68)

\begin{align*} P\uparrow\left(P\nrightarrow Q\right) & \Leftrightarrow\lnot P\lor\left(P\rightarrow Q\right)\\ & \Leftrightarrow\lnot P\lor\left(\lnot P\lor Q\right)\\ & \Leftrightarrow\left(\lnot P\lor\lnot P\right)\lor Q\\ & \Leftrightarrow P\rightarrow Q \end{align*}

(69)

\begin{align*} P\uparrow\left(P\nleftarrow Q\right) & \Leftrightarrow\lnot P\lor\left(P\leftarrow Q\right)\\ & \Leftrightarrow\lnot P\lor\left(P\lor\lnot Q\right)\\ & \Leftrightarrow\left(\lnot P\lor P\right)\lor\lnot Q\\ & \Leftrightarrow1 \end{align*}

(70)

\begin{align*} P\uparrow\left(P\nleftrightarrow Q\right) & \Leftrightarrow\lnot P\lor\left(P\leftrightarrow Q\right)\\ & \Leftrightarrow\lnot P\lor\left\{ \left(\lnot P\lor Q\right)\land\left(P\lor\lnot Q\right)\right\} \\ & \Leftrightarrow\lnot P\lor\left(\lnot P\lor Q\right)\\ & \Leftrightarrow\left(\lnot P\lor\lnot P\right)\lor Q\\ & \Leftrightarrow P\rightarrow Q \end{align*}

(71)

\begin{align*} P\nrightarrow\left(P\lor Q\right) & \Leftrightarrow P\land\lnot\left(P\lor Q\right)\\ & \Leftrightarrow P\land\left(\lnot P\land\lnot Q\right)\\ & \Leftrightarrow\left(P\land\lnot P\right)\land\lnot Q\\ & \Leftrightarrow0 \end{align*}

(72)

\begin{align*} P\nrightarrow\left(P\land Q\right) & \Leftrightarrow P\land\lnot\left(P\land Q\right)\\ & \Leftrightarrow P\land\left(\lnot P\lor\lnot Q\right)\\ & \Leftrightarrow P\land\lnot Q\\ & \Leftrightarrow P\nrightarrow Q \end{align*}

(73)

\begin{align*} P\nrightarrow\left(P\rightarrow Q\right) & \Leftrightarrow P\land\left(P\nrightarrow Q\right)\\ & \Leftrightarrow P\land\left(P\land\lnot Q\right)\\ & \Leftrightarrow\left(P\land P\right)\land\lnot Q\\ & \Leftrightarrow P\nrightarrow Q \end{align*}

(74)

\begin{align*} P\nrightarrow\left(P\leftarrow Q\right) & \Leftrightarrow P\land\left(P\nleftarrow Q\right)\\ & \Leftrightarrow P\land\left(\lnot P\land Q\right)\\ & \Leftrightarrow\left(P\land\lnot P\right)\land Q\\ & \Leftrightarrow0 \end{align*}

(75)

\begin{align*} P\nrightarrow\left(P\leftrightarrow Q\right) & \Leftrightarrow P\land\left(P\nleftrightarrow Q\right)\\ & \Leftrightarrow P\land\left\{ \left(P\land\lnot Q\right)\lor\left(\lnot P\land Q\right)\right\} \\ & \Leftrightarrow P\land\left(P\land\lnot Q\right)\\ & \Leftrightarrow\left(P\land P\right)\land\lnot Q\\ & \Leftrightarrow P\nrightarrow Q \end{align*}

(76)

\begin{align*} P\nrightarrow\left(P\downarrow Q\right) & \Leftrightarrow P\land\left(P\lor Q\right)\\ & \Leftrightarrow P \end{align*}

(77)

\begin{align*} P\nrightarrow\left(P\uparrow Q\right) & \Leftrightarrow P\land\left(P\land Q\right)\\ & \Leftrightarrow\left(P\land P\right)\land Q\\ & \Leftrightarrow P\land Q \end{align*}

(78)

\begin{align*} P\nrightarrow\left(P\nrightarrow Q\right) & \Leftrightarrow P\land\left(P\rightarrow Q\right)\\ & \Leftrightarrow P\land\left(\lnot P\lor Q\right)\\ & \Leftrightarrow P\land Q \end{align*}

(79)

\begin{align*} P\nrightarrow\left(P\nleftarrow Q\right) & \Leftrightarrow P\land\left(P\leftarrow Q\right)\\ & \Leftrightarrow P\land\left(P\lor\lnot Q\right)\\ & \Leftrightarrow P \end{align*}

(80)

\begin{align*} P\nrightarrow\left(P\nleftrightarrow Q\right) & \Leftrightarrow P\land\left(P\leftrightarrow Q\right)\\ & \Leftrightarrow P\land\left\{ \left(\lnot P\lor Q\right)\lor\left(P\lor\lnot Q\right)\right\} \\ & \Leftrightarrow P\land\left(\lnot P\lor Q\right)\\ & \Leftrightarrow P\land Q \end{align*}

(81)

\begin{align*} P\nleftarrow\left(P\lor Q\right) & \Leftrightarrow\lnot P\land\left(P\lor Q\right)\\ & \Leftrightarrow\lnot P\land Q\\ & \Leftrightarrow P\nleftarrow Q \end{align*}

(82)

\begin{align*} P\nleftarrow\left(P\land Q\right) & \Leftrightarrow\lnot P\land\left(P\land Q\right)\\ & \Leftrightarrow\left(\lnot P\land P\right)\land Q\\ & \Leftrightarrow0 \end{align*}

(83)

\begin{align*} P\nleftarrow\left(P\rightarrow Q\right) & \Leftrightarrow\lnot P\land\left(\lnot P\lor Q\right)\\ & \Leftrightarrow\lnot P \end{align*}

(84)

\begin{align*} P\nleftarrow\left(P\leftarrow Q\right) & \Leftrightarrow\lnot P\land\left(P\lor\lnot Q\right)\\ & \Leftrightarrow\lnot P\land\lnot Q\\ & \Leftrightarrow P\downarrow Q \end{align*}

(85)

\begin{align*} P\nleftarrow\left(P\leftrightarrow Q\right) & \Leftrightarrow\lnot P\land\left\{ \left(P\lor\lnot Q\right)\land\left(\lnot P\lor Q\right)\right\} \\ & \Leftrightarrow\lnot P\land\left(P\lor\lnot Q\right)\land\left(\lnot P\lor Q\right)\\ & \Leftrightarrow\lnot P\land\left(P\lor\lnot Q\right)\\ & \Leftrightarrow\lnot P\land\lnot Q\\ & \Leftrightarrow P\downarrow Q \end{align*}

(86)

\begin{align*} P\nleftarrow\left(P\downarrow Q\right) & \Leftrightarrow\lnot P\land\left(\lnot P\land\lnot Q\right)\\ & \Leftrightarrow\left(\lnot P\land\lnot P\right)\land\lnot Q\\ & \Leftrightarrow P\downarrow Q \end{align*}

(87)

\begin{align*} P\nleftarrow\left(P\uparrow Q\right) & \Leftrightarrow\lnot P\land\left(\lnot P\lor\lnot Q\right)\\ & \Leftrightarrow\lnot P \end{align*}

(88)

\begin{align*} P\nleftarrow\left(P\nrightarrow Q\right) & \Leftrightarrow\lnot P\land\left(P\land\lnot Q\right)\\ & \Leftrightarrow\left(\lnot P\land P\right)\land\lnot Q\\ & \Leftrightarrow0 \end{align*}

(89)

\begin{align*} P\nleftarrow\left(P\nleftarrow Q\right) & \Leftrightarrow\lnot P\land\left(\lnot P\land Q\right)\\ & \Leftrightarrow\left(\lnot P\land\lnot P\right)\land Q\\ & \Leftrightarrow P\nleftarrow Q \end{align*}

(90)

\begin{align*} P\nleftarrow\left(P\nleftrightarrow Q\right) & \Leftrightarrow\lnot P\land\left\{ \left(P\land\lnot Q\right)\lor\left(\lnot P\land Q\right)\right\} \\ & \Leftrightarrow\lnot P\land\left(\lnot P\land Q\right)\\ & \Leftrightarrow\left(\lnot P\land\lnot P\right)\land Q\\ & \Leftrightarrow P\nleftarrow Q \end{align*}

(91)

\begin{align*} P\nleftrightarrow\left(P\lor Q\right) & \Leftrightarrow\left\{ P\land\lnot\left(P\lor Q\right)\right\} \lor\left\{ \lnot P\land\left(P\lor Q\right)\right\} \\ & \Leftrightarrow\left\{ P\land\left(\lnot P\land\lnot Q\right)\right\} \lor\left(\lnot P\land Q\right)\\ & \Leftrightarrow\left\{ \left(P\land\lnot P\right)\land\lnot Q\right\} \lor\left(\lnot P\land Q\right)\\ & \Leftrightarrow\lnot P\land Q\\ & \Leftrightarrow P\nleftarrow Q \end{align*}

(92)

\begin{align*} P\nleftrightarrow\left(P\land Q\right) & \Leftrightarrow\left\{ P\land\lnot\left(P\land Q\right)\right\} \lor\left\{ \lnot P\land\left(P\land Q\right)\right\} \\ & \Leftrightarrow\left\{ P\land\left(\lnot P\lor\lnot Q\right)\right\} \lor\left\{ \left(\lnot P\land P\right)\land Q\right\} \\ & \Leftrightarrow P\land\left(\lnot P\lor\lnot Q\right)\\ & \Leftrightarrow P\land\lnot Q\\ & \Leftrightarrow P\nrightarrow Q \end{align*}

(93)

\begin{align*} P\nleftrightarrow\left(P\rightarrow Q\right) & \Leftrightarrow\left\{ P\land\lnot\left(P\rightarrow Q\right)\right\} \lor\left\{ \lnot P\land\left(P\rightarrow Q\right)\right\} \\ & \Leftrightarrow\left\{ P\land\left(P\land\lnot Q\right)\right\} \lor\left\{ \lnot P\land\left(\lnot P\lor Q\right)\right\} \\ & \Leftrightarrow\left\{ \left(P\land P\right)\land\lnot Q\right\} \lor\lnot P\\ & \Leftrightarrow\left\{ P\land\lnot Q\right\} \lor\lnot P\\ & \Leftrightarrow\lnot Q\lor\lnot P\\ & \Leftrightarrow P\uparrow Q \end{align*}

(94)

\begin{align*} P\nleftrightarrow\left(P\leftarrow Q\right) & \Leftrightarrow\left\{ P\land\lnot\left(P\leftarrow Q\right)\right\} \lor\left\{ \lnot P\land\left(P\leftarrow Q\right)\right\} \\ & \Leftrightarrow\left\{ P\land\left(\lnot P\land Q\right)\right\} \lor\left\{ \lnot P\land\left(P\lor\lnot Q\right)\right\} \\ & \Leftrightarrow\left\{ \left(P\land\lnot P\right)\land Q\right\} \lor\left\{ \lnot P\land\lnot Q\right\} \\ & \Leftrightarrow\lnot P\land\lnot Q\\ & \Leftrightarrow P\downarrow Q \end{align*}

(95)

\begin{align*} P\nleftrightarrow\left(P\leftrightarrow Q\right) & \Leftrightarrow\left(P\nleftrightarrow P\right)\leftrightarrow Q\\ & \Leftrightarrow0\leftrightarrow Q\\ & \Leftrightarrow\lnot Q \end{align*}

(96)

\begin{align*} P\nleftrightarrow\left(P\downarrow Q\right) & \Leftrightarrow\left\{ P\land\lnot\left(P\downarrow Q\right)\right\} \lor\left\{ \lnot P\land\left(P\downarrow Q\right)\right\} \\ & \Leftrightarrow\left\{ P\land\left(P\lor Q\right)\right\} \lor\left\{ \lnot P\land\left(\lnot P\land\lnot Q\right)\right\} \\ & \Leftrightarrow P\lor\left\{ \left(\lnot P\land\lnot P\right)\land\lnot Q\right\} \\ & \Leftrightarrow P\lor\left(\lnot P\land\lnot Q\right)\\ & \Leftrightarrow P\lor\lnot Q\\ & \Leftrightarrow P\leftarrow Q \end{align*}

(97)

\begin{align*} P\nleftrightarrow\left(P\uparrow Q\right) & \Leftrightarrow\left\{ P\land\lnot\left(P\uparrow Q\right)\right\} \lor\left\{ \lnot P\land\left(P\uparrow Q\right)\right\} \\ & \Leftrightarrow\left\{ P\land\left(P\land Q\right)\right\} \lor\left\{ \lnot P\land\left(\lnot P\lor\lnot Q\right)\right\} \\ & \Leftrightarrow\left\{ \left(P\land P\right)\land Q\right\} \lor\lnot P\\ & \Leftrightarrow\left(P\land Q\right)\lor\lnot P\\ & \Leftrightarrow Q\lor\lnot P\\ & \Leftrightarrow P\rightarrow Q \end{align*}

(98)

\begin{align*} P\nleftrightarrow\left(P\nrightarrow Q\right) & \Leftrightarrow\left\{ P\land\lnot\left(P\nrightarrow Q\right)\right\} \lor\left\{ \lnot P\land\left(P\nrightarrow Q\right)\right\} \\ & \Leftrightarrow\left\{ P\land\left(\lnot P\lor Q\right)\right\} \lor\left\{ \lnot P\land\left(P\land\lnot Q\right)\right\} \\ & \Leftrightarrow\left\{ P\land Q\right\} \lor\left\{ \left(\lnot P\land P\right)\land\lnot Q\right\} \\ & \Leftrightarrow P\land Q \end{align*}

(99)

\begin{align*} P\nleftrightarrow\left(P\nleftarrow Q\right) & \Leftrightarrow\left\{ P\land\lnot\left(P\nleftarrow Q\right)\right\} \lor\left\{ \lnot P\land\left(P\nleftarrow Q\right)\right\} \\ & \Leftrightarrow\left\{ P\land\left(P\lor\lnot Q\right)\right\} \lor\left\{ \lnot P\land\left(\lnot P\land Q\right)\right\} \\ & \Leftrightarrow P\lor\left\{ \left(\lnot P\land\lnot P\right)\land Q\right\} \\ & \Leftrightarrow P\lor\left(\lnot P\land Q\right)\\ & \Leftrightarrow P\lor Q \end{align*}

(100)

\begin{align*} P\nleftrightarrow\left(P\nleftrightarrow Q\right) & \Leftrightarrow\left(P\nleftrightarrow P\right)\nleftrightarrow Q\\ & \Leftrightarrow0\nleftrightarrow Q\\ & \Leftrightarrow Q \end{align*}

(101)

\begin{align*} P\lor\left(\lnot P\lor Q\right) & \Leftrightarrow P\lor\lnot P\lor Q\\ & \Leftrightarrow1 \end{align*}

(102)

\begin{align*} P\lor\left(\lnot P\land Q\right) & \Leftrightarrow\left(P\lor\lnot P\right)\land\left(P\lor Q\right)\\ & \Leftrightarrow P\lor Q \end{align*}

(103)

\begin{align*} P\lor\left(\lnot P\rightarrow Q\right) & \Leftrightarrow P\lor\left(P\lor Q\right)\\ & \Leftrightarrow P\lor P\lor Q\\ & \Leftrightarrow P\lor Q \end{align*}

(104)

\begin{align*} P\lor\left(\lnot P\leftarrow Q\right) & \Leftrightarrow P\lor\lnot P\lor\lnot Q\\ & \Leftrightarrow1 \end{align*}

(105)

\begin{align*} P\lor\left(\lnot P\leftrightarrow Q\right) & \Leftrightarrow P\lor\left\{ \left(P\lor Q\right)\land\left(\lnot P\lor\lnot Q\right)\right\} \\ & \Leftrightarrow P\lor P\lor Q\\ & \Leftrightarrow P\lor Q \end{align*}

(106)

\begin{align*} P\lor\left(\lnot P\downarrow Q\right) & \Leftrightarrow P\lor\left(P\land\lnot Q\right)\\ & \Leftrightarrow P \end{align*}

(107)

\begin{align*} P\lor\left(\lnot P\uparrow Q\right) & \Leftrightarrow P\lor P\lor\lnot Q\\ & \Leftrightarrow P\leftarrow Q \end{align*}

(108)

\begin{align*} P\lor\left(\lnot P\nrightarrow Q\right) & \Leftrightarrow P\lor\left(\lnot P\land\lnot Q\right)\\ & \Leftrightarrow P\lor\lnot Q\\ & \Leftrightarrow P\leftarrow Q \end{align*}

(109)

\begin{align*} P\lor\left(\lnot P\nleftarrow Q\right) & P\lor\left(P\land Q\right)\\ & \Leftrightarrow P \end{align*}

(110)

\begin{align*} P\lor\left(\lnot P\nleftrightarrow Q\right) & \Leftrightarrow P\lor\left(\lnot P\land\lnot Q\right)\lor\left(P\land Q\right)\\ & \Leftrightarrow P\lor\left(\lnot P\land\lnot Q\right)\\ & \Leftrightarrow P\lor\lnot Q\\ & \Leftrightarrow P\leftarrow Q \end{align*}

(111)

\begin{align*} P\land\left(\lnot P\lor Q\right) & \Leftrightarrow\left(P\land\lnot P\right)\lor\left(P\land Q\right)\\ & \Leftrightarrow P\land Q \end{align*}

(112)

\begin{align*} P\land\left(\lnot P\land Q\right) & \Leftrightarrow P\land\lnot P\land Q\\ & \Leftrightarrow0 \end{align*}

(113)

\begin{align*} P\land\left(\lnot P\rightarrow Q\right) & \Leftrightarrow P\land\left(P\lor Q\right)\\ & \Leftrightarrow P \end{align*}

(114)

\begin{align*} P\land\left(\lnot P\leftarrow Q\right) & \Leftrightarrow P\land\left(\lnot P\lor\lnot Q\right)\\ & \Leftrightarrow P\land\lnot Q\\ & \Leftrightarrow P\nrightarrow Q \end{align*}

(115)

\begin{align*} P\land\left(\lnot P\leftrightarrow Q\right) & \Leftrightarrow P\land\left(P\lor Q\right)\land\left(\lnot P\lor\lnot Q\right)\\ & \Leftrightarrow P\land\left(\lnot P\lor\lnot Q\right)\\ & \Leftrightarrow P\land\lnot Q\\ & \Leftrightarrow P\nrightarrow Q \end{align*}

(116)

\begin{align*} P\land\left(\lnot P\downarrow Q\right) & \Leftrightarrow P\land P\land\lnot Q\\ & \Leftrightarrow P\land\lnot Q\\ & \Leftrightarrow P\nrightarrow Q \end{align*}

(117)

\begin{align*} P\land\left(\lnot P\uparrow Q\right) & \Leftrightarrow P\land\left(P\lor\lnot Q\right)\\ & \Leftrightarrow P \end{align*}

(118)

\begin{align*} P\land\left(\lnot P\nrightarrow Q\right) & \Leftrightarrow P\land\lnot P\land\lnot Q\\ & \Leftrightarrow0 \end{align*}

(119)

\begin{align*} P\land\left(\lnot P\nleftarrow Q\right) & \Leftrightarrow P\land P\land Q\\ & \Leftrightarrow P\land Q \end{align*}

(120)

\begin{align*} P\land\left(\lnot P\nleftrightarrow Q\right) & \Leftrightarrow P\land\left\{ \left(\lnot P\land\lnot Q\right)\lor\left(P\land Q\right)\right\} \\ & \Leftrightarrow P\land P\land Q\\ & \Leftrightarrow P\land Q \end{align*}

(121)

\begin{align*} P\rightarrow\left(\lnot P\lor Q\right) & \Leftrightarrow\lnot P\lor\lnot P\lor Q\\ & \Leftrightarrow\lnot P\lor Q\\ & \Leftrightarrow P\rightarrow Q \end{align*}

(122)

\begin{align*} P\rightarrow\left(\lnot P\land Q\right) & \Leftrightarrow\lnot P\lor\left(\lnot P\land Q\right)\\ & \Leftrightarrow\lnot P \end{align*}

(123)

\begin{align*} P\rightarrow\left(\lnot P\rightarrow Q\right) & \Leftrightarrow\lnot P\lor P\lor Q\\ & \Leftrightarrow1 \end{align*}

(124)

\begin{align*} P\rightarrow\left(\lnot P\leftarrow Q\right) & \Leftrightarrow\lnot P\lor\lnot P\lor\lnot Q\\ & \Leftrightarrow\lnot P\lor\lnot Q\\ & \Leftrightarrow P\uparrow Q \end{align*}

(125)

\begin{align*} P\rightarrow\left(\lnot P\leftrightarrow Q\right) & \Leftrightarrow\lnot P\lor\left\{ \left(P\lor Q\right)\land\left(\lnot P\lor\lnot Q\right)\right\} \\ & \Leftrightarrow\lnot P\lor\left\{ \left(P\lor Q\right)\land\left(\lnot P\lor\lnot Q\right)\right\} \\ & \Leftrightarrow\lnot P\lor\lnot P\lor\lnot Q\\ & \Leftrightarrow\lnot P\lor\lnot Q\\ & \Leftrightarrow P\uparrow Q \end{align*}

(126)

\begin{align*} P\rightarrow\left(\lnot P\downarrow Q\right) & \Leftrightarrow\lnot P\lor\left(P\land\lnot Q\right)\\ & \Leftrightarrow\lnot P\lor\lnot Q\\ & \Leftrightarrow P\uparrow Q \end{align*}

(127)

\begin{align*} P\rightarrow\left(\lnot P\uparrow Q\right) & \Leftrightarrow\lnot P\lor P\lor\lnot Q\\ & \Leftrightarrow1 \end{align*}

(128)

\begin{align*} P\rightarrow\left(\lnot P\nrightarrow Q\right) & \Leftrightarrow\lnot P\lor\left(\lnot P\land\lnot Q\right)\\ & \Leftrightarrow\lnot P \end{align*}

(129)

\begin{align*} P\rightarrow\left(\lnot P\nleftarrow Q\right) & \Leftrightarrow\lnot P\lor\left(P\land Q\right)\\ & \Leftrightarrow\lnot P\lor Q\\ & \Leftrightarrow P\rightarrow Q \end{align*}

(130)

\begin{align*} P\rightarrow\left(\lnot P\nleftrightarrow Q\right) & \Leftrightarrow\lnot P\lor\left(\lnot P\land\lnot Q\right)\lor\left(P\land Q\right)\\ & \Leftrightarrow\lnot P\lor\left(P\land Q\right)\\ & \Leftrightarrow\lnot P\lor Q\\ & \Leftrightarrow P\rightarrow Q \end{align*}

(131)

\begin{align*} P\leftarrow\left(\lnot P\lor Q\right) & \Leftrightarrow P\lor\lnot\left(\lnot P\lor Q\right)\\ & \Leftrightarrow P\lor\left(P\land\lnot Q\right)\\ & \Leftrightarrow P \end{align*}

(132)

\begin{align*} P\leftarrow\left(\lnot P\land Q\right) & \Leftrightarrow P\lor\lnot\left(\lnot P\land Q\right)\\ & \Leftrightarrow P\lor P\lor\lnot Q\\ & \Leftrightarrow P\lor\lnot Q\\ & \Leftrightarrow P\leftarrow Q \end{align*}

(133)

\begin{align*} P\leftarrow\left(\lnot P\rightarrow Q\right) & \Leftrightarrow P\lor\left(\lnot P\nrightarrow Q\right)\\ & \Leftrightarrow P\lor\left(\lnot P\land\lnot Q\right)\\ & \Leftrightarrow P\lor\lnot Q\\ & \Leftrightarrow P\leftarrow Q \end{align*}

(134)

\begin{align*} P\leftarrow\left(\lnot P\leftarrow Q\right) & \Leftrightarrow P\lor\left(\lnot P\nleftarrow Q\right)\\ & \Leftrightarrow P\lor\left(P\land Q\right)\\ & \Leftrightarrow P \end{align*}

(135)

\begin{align*} P\leftarrow\left(\lnot P\leftrightarrow Q\right) & \Leftrightarrow P\lor\left(\lnot P\nleftrightarrow Q\right)\\ & \Leftrightarrow P\lor\left(\lnot P\land\lnot Q\right)\lor\left(P\land Q\right)\\ & \Leftrightarrow P\lor\left(\lnot P\land\lnot Q\right)\\ & \Leftrightarrow P\lor\lnot Q\\ & \Leftrightarrow P\leftarrow Q \end{align*}

(136)

\begin{align*} P\leftarrow\left(\lnot P\downarrow Q\right) & \Leftrightarrow P\lor\lnot\left(\lnot P\downarrow Q\right)\\ & \Leftrightarrow P\lor\lnot P\lor Q\\ & \Leftrightarrow1 \end{align*}

(137)

\begin{align*} P\leftarrow\left(\lnot P\uparrow Q\right) & \Leftrightarrow P\lor\lnot\left(\lnot P\uparrow Q\right)\\ & \Leftrightarrow P\lor\left(\lnot P\land Q\right)\\ & \Leftrightarrow P\lor Q \end{align*}

(138)

\begin{align*} P\leftarrow\left(\lnot P\nrightarrow Q\right) & \Leftrightarrow P\lor\left(\lnot P\rightarrow Q\right)\\ & \Leftrightarrow P\lor P\lor Q\\ & \Leftrightarrow P\lor Q \end{align*}

(139)

\begin{align*} P\leftarrow\left(\lnot P\nleftarrow Q\right) & \Leftrightarrow P\lor\left(\lnot P\leftarrow Q\right)\\ & \Leftrightarrow P\lor\lnot P\lor\lnot Q\\ & \Leftrightarrow1 \end{align*}

(140)

\begin{align*} P\leftarrow\left(\lnot P\nleftrightarrow Q\right) & \Leftrightarrow P\lor\left(\lnot P\leftrightarrow Q\right)\\ & \Leftrightarrow P\lor\left\{ \left(P\lor Q\right)\land\left(\lnot P\lor\lnot Q\right)\right\} \\ & \Leftrightarrow P\lor P\lor Q\\ & \Leftrightarrow P\lor Q \end{align*}

(141)

\begin{align*} P\leftrightarrow\left(\lnot P\lor Q\right) & \Leftrightarrow\left\{ \lnot P\lor\lnot P\lor Q\right\} \land\left\{ P\lor\lnot\left(\lnot P\lor Q\right)\right\} \\ & \Leftrightarrow\left\{ \lnot P\lor Q\right\} \land\left\{ P\lor\left(P\land\lnot Q\right)\right\} \\ & \Leftrightarrow\left\{ \lnot P\lor Q\right\} \land P\\ & \Leftrightarrow P\land Q \end{align*}

(142)

\begin{align*} P\leftrightarrow\left(\lnot P\land Q\right) & \Leftrightarrow\left\{ \lnot P\lor\left(\lnot P\land Q\right)\right\} \land\left\{ P\lor\lnot\left(\lnot P\land Q\right)\right\} \\ & \Leftrightarrow\lnot P\land\left\{ P\lor P\lor\lnot Q\right\} \\ & \Leftrightarrow\lnot P\land\left\{ P\lor\lnot Q\right\} \\ & \Leftrightarrow\lnot P\land\lnot Q\\ & \Leftrightarrow P\downarrow Q \end{align*}

(143)

\begin{align*} P\leftrightarrow\left(\lnot P\rightarrow Q\right) & \Leftrightarrow\left\{ \lnot P\lor\left(\lnot P\rightarrow Q\right)\right\} \land\left\{ P\lor\left(\lnot P\nrightarrow Q\right)\right\} \\ & \Leftrightarrow\left\{ \lnot P\lor P\lor Q\right\} \land\left\{ P\lor\left(\lnot P\land\lnot Q\right)\right\} \\ & \Leftrightarrow P\lor\left(\lnot P\land\lnot Q\right)\\ & \Leftrightarrow P\lor\lnot Q\\ & \Leftrightarrow P\leftarrow Q \end{align*}

(144)

\begin{align*} P\leftrightarrow\left(\lnot P\leftarrow Q\right) & \Leftrightarrow\left\{ \lnot P\lor\left(\lnot P\leftarrow Q\right)\right\} \land\left\{ P\lor\left(\lnot P\nleftarrow Q\right)\right\} \\ & \Leftrightarrow\left\{ \lnot P\lor\lnot P\lor\lnot Q\right\} \land\left\{ P\lor\left(P\land Q\right)\right\} \\ & \Leftrightarrow\left\{ \lnot P\lor\lnot Q\right\} \land P\\ & \Leftrightarrow\lnot Q\land P\\ & \Leftrightarrow P\nrightarrow Q \end{align*}

(145)

\begin{align*} P\leftrightarrow\left(\lnot P\leftrightarrow Q\right) & \Leftrightarrow P\leftrightarrow\lnot P\leftrightarrow Q\\ & \Leftrightarrow0\leftrightarrow Q\\ & \Leftrightarrow\lnot Q \end{align*}

(146)

\begin{align*} P\leftrightarrow\left(\lnot P\downarrow Q\right) & \Leftrightarrow\left\{ \lnot P\lor\left(\lnot P\downarrow Q\right)\right\} \land\left\{ P\lor\lnot\left(\lnot P\downarrow Q\right)\right\} \\ & \Leftrightarrow\left\{ \lnot P\lor\left(P\land\lnot Q\right)\right\} \land\left\{ P\lor\lnot P\lor Q\right\} \\ & \Leftrightarrow\lnot P\lor\left(P\land\lnot Q\right)\\ & \Leftrightarrow\lnot P\lor\lnot Q\\ & \Leftrightarrow P\uparrow Q \end{align*}

(147)

\begin{align*} P\leftrightarrow\left(\lnot P\uparrow Q\right) & \Leftrightarrow\left\{ \lnot P\lor\left(\lnot P\uparrow Q\right)\right\} \land\left\{ P\lor\lnot\left(\lnot P\uparrow Q\right)\right\} \\ & \Leftrightarrow\left\{ \lnot P\lor P\lor\lnot Q\right\} \land\left\{ P\lor\left(\lnot P\land Q\right)\right\} \\ & \Leftrightarrow P\lor\left(\lnot P\land Q\right)\\ & \Leftrightarrow P\lor Q \end{align*}

(148)

\begin{align*} P\leftrightarrow\left(\lnot P\nrightarrow Q\right) & \Leftrightarrow\left\{ \lnot P\lor\left(\lnot P\nrightarrow Q\right)\right\} \land\left\{ P\lor\left(\lnot P\rightarrow Q\right)\right\} \\ & \Leftrightarrow\left\{ \lnot P\lor\left(\lnot P\land\lnot Q\right)\right\} \land\left\{ P\lor P\lor Q\right\} \\ & \Leftrightarrow\lnot P\land\left\{ P\lor Q\right\} \\ & \Leftrightarrow\lnot P\land Q\\ & \Leftrightarrow P\nleftarrow Q \end{align*}

(149)

\begin{align*} P\leftrightarrow\left(\lnot P\nleftarrow Q\right) & \Leftrightarrow\left\{ \lnot P\lor\left(\lnot P\nleftarrow Q\right)\right\} \land\left\{ P\lor\left(\lnot P\leftarrow Q\right)\right\} \\ & \Leftrightarrow\left\{ \lnot P\lor\left(P\land Q\right)\right\} \land\left\{ P\lor\lnot P\lor\lnot Q\right\} \\ & \Leftrightarrow\lnot P\lor Q\\ & \Leftrightarrow P\rightarrow Q \end{align*}

(150)

\begin{align*} P\leftrightarrow\left(\lnot P\nleftrightarrow Q\right) & \Leftrightarrow P\leftrightarrow\lnot P\nleftrightarrow Q\\ & \Leftrightarrow0\nleftrightarrow Q\\ & \Leftrightarrow Q \end{align*}

(151)

\begin{align*} P\downarrow\left(\lnot P\lor Q\right) & \Leftrightarrow\lnot P\land\lnot\left(\lnot P\lor Q\right)\\ & \Leftrightarrow\lnot P\land P\land\lnot Q\\ & \Leftrightarrow0 \end{align*}

(152)

\begin{align*} P\downarrow\left(\lnot P\land Q\right) & \Leftrightarrow\lnot P\land\lnot\left(\lnot P\land Q\right)\\ & \Leftrightarrow\lnot P\land\left(P\lor\lnot Q\right)\\ & \Leftrightarrow\lnot P\land\lnot Q\\ & \Leftrightarrow P\downarrow Q \end{align*}

(153)

\begin{align*} P\downarrow\left(\lnot P\rightarrow Q\right) & \Leftrightarrow\lnot P\land\left(\lnot P\nrightarrow Q\right)\\ & \Leftrightarrow\lnot P\land\lnot P\land\lnot Q\\ & \Leftrightarrow\lnot P\land\lnot Q\\ & \Leftrightarrow P\downarrow Q \end{align*}

(154)

\begin{align*} P\downarrow\left(\lnot P\leftarrow Q\right) & \Leftrightarrow\lnot P\land\left(\lnot P\nleftarrow Q\right)\\ & \Leftrightarrow\lnot P\land P\land Q\\ & \Leftrightarrow0 \end{align*}

(155)

\begin{align*} P\downarrow\left(\lnot P\leftrightarrow Q\right) & \Leftrightarrow\lnot P\land\left(\lnot P\nleftrightarrow Q\right)\\ & \Leftrightarrow\lnot P\land\left\{ \left(\lnot P\land\lnot Q\right)\lor\left(P\land Q\right)\right\} \\ & \Leftrightarrow\lnot P\land\left(\lnot P\land\lnot Q\right)\\ & \Leftrightarrow\lnot P\land\lnot Q\\ & \Leftrightarrow P\downarrow Q \end{align*}

(156)

\begin{align*} P\downarrow\left(\lnot P\downarrow Q\right) & \Leftrightarrow\lnot P\land\lnot\left(\lnot P\downarrow Q\right)\\ & \Leftrightarrow\lnot P\land\left(\lnot P\lor Q\right)\\ & \Leftrightarrow\lnot P \end{align*}

(157)

\begin{align*} P\downarrow\left(\lnot P\uparrow Q\right) & \Leftrightarrow\lnot P\land\lnot\left(\lnot P\uparrow Q\right)\\ & \Leftrightarrow\lnot P\land\lnot P\land Q\\ & \Leftrightarrow\lnot P\land Q \end{align*}

(158)

\begin{align*} P\downarrow\left(\lnot P\nrightarrow Q\right) & \Leftrightarrow\lnot P\land\left(\lnot P\rightarrow Q\right)\\ & \Leftrightarrow\lnot P\land\left(P\lor Q\right)\\ & \Leftrightarrow\lnot P\land Q\\ & \Leftrightarrow P\nleftarrow Q \end{align*}

(159)

\begin{align*} P\downarrow\left(\lnot P\nleftarrow Q\right) & \Leftrightarrow\lnot P\land\left(\lnot P\leftarrow Q\right)\\ & \Leftrightarrow\lnot P\land\left(\lnot P\lor\lnot Q\right)\\ & \Leftrightarrow\lnot P \end{align*}

(160)

\begin{align*} P\downarrow\left(\lnot P\nleftrightarrow Q\right) & \Leftrightarrow\lnot P\land\left(\lnot P\leftrightarrow Q\right)\\ & \Leftrightarrow\lnot P\land\left(P\lor Q\right)\land\left(\lnot P\lor\lnot Q\right)\\ & \Leftrightarrow\lnot P\land\left(P\lor Q\right)\\ & \Leftrightarrow\lnot P\land Q\\ & \Leftrightarrow P\nleftarrow Q \end{align*}

(161)

\begin{align*} P\uparrow\left(\lnot P\lor Q\right) & \Leftrightarrow\lnot P\lor\lnot\left(\lnot P\lor Q\right)\\ & \Leftrightarrow\lnot P\lor\left(P\land\lnot Q\right)\\ & \Leftrightarrow\lnot P\lor\lnot Q\\ & \Leftrightarrow P\uparrow Q \end{align*}

(162)

\begin{align*} P\uparrow\left(\lnot P\land Q\right) & \Leftrightarrow\lnot P\lor\lnot\left(\lnot P\land Q\right)\\ & \Leftrightarrow\lnot P\lor P\lor\lnot Q\\ & \Leftrightarrow1 \end{align*}

(163)

\begin{align*} P\uparrow\left(\lnot P\rightarrow Q\right) & \Leftrightarrow\lnot P\lor\left(\lnot P\nrightarrow Q\right)\\ & \Leftrightarrow\lnot P\lor\left(\lnot P\land\lnot Q\right)\\ & \Leftrightarrow\lnot P \end{align*}

(164)

\begin{align*} P\uparrow\left(\lnot P\leftarrow Q\right) & \Leftrightarrow\lnot P\lor\left(\lnot P\nleftarrow Q\right)\\ & \Leftrightarrow\lnot P\lor\left(P\land Q\right)\\ & \Leftrightarrow\lnot P\lor Q\\ & \Leftrightarrow P\rightarrow Q \end{align*}

(165)

\begin{align*} P\uparrow\left(\lnot P\leftrightarrow Q\right) & \Leftrightarrow\lnot P\lor\left(\lnot P\nleftrightarrow Q\right)\\ & \Leftrightarrow\lnot P\lor\left(\lnot P\land\lnot Q\right)\lor\left(P\land Q\right)\\ & \Leftrightarrow\lnot P\lor\left(P\land Q\right)\\ & \Leftrightarrow\lnot P\lor Q\\ & \Leftrightarrow P\rightarrow Q \end{align*}

(166)

\begin{align*} P\uparrow\left(\lnot P\downarrow Q\right) & \Leftrightarrow\lnot P\lor\lnot\left(\lnot P\downarrow Q\right)\\ & \Leftrightarrow\lnot P\lor\lnot P\lor Q\\ & \Leftrightarrow P\rightarrow Q \end{align*}

(167)

\begin{align*} P\uparrow\left(\lnot P\uparrow Q\right) & \Leftrightarrow\lnot P\lor\lnot\left(\lnot P\uparrow Q\right)\\ & \Leftrightarrow\lnot P\lor\left(\lnot P\land Q\right)\\ & \Leftrightarrow\lnot P \end{align*}

(168)

\begin{align*} P\uparrow\left(\lnot P\nrightarrow Q\right) & \Leftrightarrow\lnot P\lor\left(\lnot P\rightarrow Q\right)\\ & \Leftrightarrow\lnot P\lor P\lor Q\\ & \Leftrightarrow1 \end{align*}

(169)

\begin{align*} P\uparrow\left(\lnot P\nleftarrow Q\right) & \Leftrightarrow\lnot P\lor\left(\lnot P\leftarrow Q\right)\\ & \Leftrightarrow\lnot P\lor\lnot P\lor\lnot Q\\ & \Leftrightarrow P\uparrow Q \end{align*}

(170)

\begin{align*} P\uparrow\left(\lnot P\nleftrightarrow Q\right) & \Leftrightarrow\lnot P\lor\left(\lnot P\leftrightarrow Q\right)\\ & \Leftrightarrow\lnot P\lor\left\{ \left(P\lor Q\right)\land\left(\lnot P\lor\lnot Q\right)\right\} \\ & \Leftrightarrow\lnot P\lor\lnot P\lor\lnot Q\\ & \Leftrightarrow P\uparrow Q \end{align*}

(171)

\begin{align*} P\nrightarrow\left(\lnot P\lor Q\right) & \Leftrightarrow P\land\lnot\left(\lnot P\lor Q\right)\\ & \Leftrightarrow P\land P\land\lnot Q\\ & \Leftrightarrow P\nrightarrow Q \end{align*}

(172)

\begin{align*} P\nrightarrow\left(\lnot P\land Q\right) & \Leftrightarrow P\land\lnot\left(\lnot P\land Q\right)\\ & \Leftrightarrow P\land\left(P\lor\lnot Q\right)\\ & \Leftrightarrow P \end{align*}

(173)

\begin{align*} P\nrightarrow\left(\lnot P\rightarrow Q\right) & \Leftrightarrow P\land\left(\lnot P\nrightarrow Q\right)\\ & \Leftrightarrow P\land\lnot P\land\lnot Q\\ & \Leftrightarrow0 \end{align*}

(174)

\begin{align*} P\nrightarrow\left(\lnot P\leftarrow Q\right) & \Leftrightarrow P\land\left(\lnot P\nleftarrow Q\right)\\ & \Leftrightarrow P\land P\land Q\\ & \Leftrightarrow P\land Q \end{align*}

(175)

\begin{align*} P\nrightarrow\left(\lnot P\leftrightarrow Q\right) & \Leftrightarrow P\land\left(\lnot P\nleftrightarrow Q\right)\\ & \Leftrightarrow P\land\left\{ \left(\lnot P\land\lnot Q\right)\lor\left(P\land Q\right)\right\} \\ & \Leftrightarrow P\land P\land Q\\ & \Leftrightarrow P\land Q \end{align*}

(176)

\begin{align*} P\nrightarrow\left(\lnot P\downarrow Q\right) & \Leftrightarrow P\land\left(\lnot P\lor Q\right)\\ & \Leftrightarrow P\land Q \end{align*}

(177)

\begin{align*} P\nrightarrow\left(\lnot P\uparrow Q\right) & \Leftrightarrow P\land\lnot P\land Q\\ & \Leftrightarrow0 \end{align*}

(178)

\begin{align*} P\nrightarrow\left(\lnot P\nrightarrow Q\right) & \Leftrightarrow P\land\left(\lnot P\rightarrow Q\right)\\ & \Leftrightarrow P\land\left(P\lor Q\right)\\ & \Leftrightarrow P \end{align*}

(179)

\begin{align*} P\nrightarrow\left(\lnot P\nleftarrow Q\right) & \Leftrightarrow P\land\left(\lnot P\leftarrow Q\right)\\ & \Leftrightarrow P\land\left(\lnot P\lor\lnot Q\right)\\ & \Leftrightarrow P\land\lnot Q\\ & \Leftrightarrow P\nrightarrow Q \end{align*}

(180)

\begin{align*} P\nrightarrow\left(\lnot P\nleftrightarrow Q\right) & \Leftrightarrow P\land\left(\lnot P\leftrightarrow Q\right)\\ & \Leftrightarrow P\land\left(P\lor Q\right)\land\left(\lnot P\lor\lnot Q\right)\\ & \Leftrightarrow P\land\left(\lnot P\lor\lnot Q\right)\\ & \Leftrightarrow P\land\lnot Q\\ & \Leftrightarrow P\nrightarrow Q \end{align*}

(181)

\begin{align*} P\nleftarrow\left(\lnot P\lor Q\right) & \Leftrightarrow\lnot P\land\left(\lnot P\lor Q\right)\\ & \Leftrightarrow\lnot P \end{align*}

(182)

\begin{align*} P\nleftarrow\left(\lnot P\land Q\right) & \Leftrightarrow\lnot P\land\lnot P\land Q\\ & \Leftrightarrow P\nleftarrow Q \end{align*}

(183)

\begin{align*} P\nleftarrow\left(\lnot P\rightarrow Q\right) & \Leftrightarrow\lnot P\land\left(\lnot P\rightarrow Q\right)\\ & \Leftrightarrow\lnot P\land\left(P\lor Q\right)\\ & \Leftrightarrow\lnot P\land Q\\ & \Leftrightarrow P\nleftarrow Q \end{align*}

(184)

\begin{align*} P\nleftarrow\left(\lnot P\leftarrow Q\right) & \Leftrightarrow\lnot P\land\left(\lnot P\leftarrow Q\right)\\ & \Leftrightarrow\lnot P\land\left(\lnot P\lor\lnot Q\right)\\ & \Leftrightarrow\lnot P \end{align*}

(185)

\begin{align*} P\nleftarrow\left(\lnot P\leftrightarrow Q\right) & \Leftrightarrow\lnot P\land\left(\lnot P\leftrightarrow Q\right)\\ & \Leftrightarrow\lnot P\land\left(P\lor Q\right)\land\left(\lnot P\lor\lnot Q\right)\\ & \Leftrightarrow\lnot P\land\left(P\lor Q\right)\\ & \Leftrightarrow\lnot P\land Q\\ & \Leftrightarrow P\nleftarrow Q \end{align*}

(186)

\begin{align*} P\nleftarrow\left(\lnot P\downarrow Q\right) & \Leftrightarrow\lnot P\land\left(\lnot P\downarrow Q\right)\\ & \Leftrightarrow\lnot P\land P\land\lnot Q\\ & \Leftrightarrow0 \end{align*}

(187)

\begin{align*} P\nleftarrow\left(\lnot P\uparrow Q\right) & \Leftrightarrow\lnot P\land\left(P\lor\lnot Q\right)\\ & \Leftrightarrow\lnot P\land\lnot Q\\ & \Leftrightarrow P\downarrow Q \end{align*}

(188)

\begin{align*} P\nleftarrow\left(\lnot P\nrightarrow Q\right) & \Leftrightarrow\lnot P\land\left(\lnot P\nrightarrow Q\right)\\ & \Leftrightarrow\lnot P\land\lnot P\land\lnot Q\\ & \Leftrightarrow P\downarrow Q \end{align*}

(189)

\begin{align*} P\nleftarrow\left(\lnot P\nleftarrow Q\right) & \Leftrightarrow\lnot P\land\left(\lnot P\nleftarrow Q\right)\\ & \Leftrightarrow\lnot P\land P\land Q\\ & \Leftrightarrow0 \end{align*}

(190)

\begin{align*} P\nleftarrow\left(\lnot P\nleftrightarrow Q\right) & \Leftrightarrow\lnot P\land\left(\lnot P\nleftrightarrow Q\right)\\ & \Leftrightarrow\lnot P\land\left\{ \left(\lnot P\land\lnot Q\right)\lor\left(P\land Q\right)\right\} \\ & \Leftrightarrow\lnot P\land\lnot P\land\lnot Q\\ & \Leftrightarrow P\downarrow Q \end{align*}

(191)

\begin{align*} P\nleftrightarrow\left(\lnot P\lor Q\right) & \Leftrightarrow\left\{ P\land\lnot\left(\lnot P\lor Q\right)\right\} \lor\left\{ \lnot P\land\left(\lnot P\lor Q\right)\right\} \\ & \Leftrightarrow\left(P\land P\land\lnot Q\right)\lor\lnot P\\ & \Leftrightarrow\left(P\land\lnot Q\right)\lor\lnot P\\ & \Leftrightarrow\lnot Q\lor\lnot P\\ & \Leftrightarrow P\uparrow Q \end{align*}

(192)

\begin{align*} P\nleftrightarrow\left(\lnot P\land Q\right) & \Leftrightarrow\left\{ P\land\lnot\left(\lnot P\land Q\right)\right\} \lor\left\{ \lnot P\land\lnot P\land Q\right\} \\ & \Leftrightarrow\left\{ P\land\left(P\lor\lnot Q\right)\right\} \lor\left(\lnot P\land Q\right)\\ & \Leftrightarrow P\lor\left(\lnot P\land Q\right)\\ & \Leftrightarrow P\lor Q \end{align*}

(193)

\begin{align*} P\nleftrightarrow\left(\lnot P\rightarrow Q\right) & \Leftrightarrow\left\{ P\land\left(\lnot P\nrightarrow Q\right)\right\} \lor\left\{ \lnot P\land\left(\lnot P\rightarrow Q\right)\right\} \\ & \Leftrightarrow\left(P\land\lnot P\land\lnot Q\right)\lor\left\{ \lnot P\land\left(P\lor Q\right)\right\} \\ & \Leftrightarrow\lnot P\land\left(P\lor Q\right)\\ & \Leftrightarrow\lnot P\land Q\\ & \Leftrightarrow P\nleftarrow Q \end{align*}

(194)

\begin{align*} P\nleftrightarrow\left(\lnot P\leftarrow Q\right) & \Leftrightarrow\left\{ P\land\left(\lnot P\nleftarrow Q\right)\right\} \lor\left\{ \lnot P\land\left(\lnot P\leftarrow Q\right)\right\} \\ & \Leftrightarrow\left(P\land P\land Q\right)\lor\left\{ \lnot P\land\left(\lnot P\lor\lnot Q\right)\right\} \\ & \Leftrightarrow\left(P\land Q\right)\lor\lnot P\\ & \Leftrightarrow Q\lor\lnot P\\ & \Leftrightarrow P\rightarrow Q \end{align*}

(195)

\begin{align*} P\nleftrightarrow\left(\lnot P\leftrightarrow Q\right) & \Leftrightarrow\left(P\nleftrightarrow\lnot P\right)\leftrightarrow Q\\ & \Leftrightarrow\left(P\leftrightarrow P\right)\leftrightarrow Q\\ & \Leftrightarrow1\leftrightarrow Q\\ & \Leftrightarrow Q \end{align*}

(196)

\begin{align*} P\nleftrightarrow\left(\lnot P\downarrow Q\right) & \Leftrightarrow\left\{ P\land\left(\lnot P\lor Q\right)\right\} \lor\left\{ \lnot P\land\left(\lnot P\downarrow Q\right)\right\} \\ & \Leftrightarrow\left(P\land Q\right)\lor\left(\lnot P\land P\land\lnot Q\right)\\ & \Leftrightarrow P\land Q \end{align*}

(197)

\begin{align*} P\nleftrightarrow\left(\lnot P\uparrow Q\right) & \Leftrightarrow\left\{ P\land\lnot P\land Q\right\} \lor\left\{ \lnot P\land\left(\lnot P\uparrow Q\right)\right\} \\ & \Leftrightarrow\lnot P\land\left(P\lor\lnot Q\right)\\ & \Leftrightarrow\lnot P\land\lnot Q\\ & \Leftrightarrow P\downarrow Q \end{align*}

(198)

\begin{align*} P\nleftrightarrow\left(\lnot P\nrightarrow Q\right) & \Leftrightarrow\left\{ P\land\left(\lnot P\rightarrow Q\right)\right\} \lor\left\{ \lnot P\land\left(\lnot P\nrightarrow Q\right)\right\} \\ & \Leftrightarrow\left\{ P\land\left(P\lor Q\right)\right\} \lor\left\{ \lnot P\land\lnot P\land\lnot Q\right\} \\ & \Leftrightarrow P\lor\left(\lnot P\land\lnot Q\right)\\ & \Leftrightarrow P\lor\lnot Q\\ & \Leftrightarrow P\leftarrow Q \end{align*}

(199)

\begin{align*} P\nleftrightarrow\left(\lnot P\nleftarrow Q\right) & \Leftrightarrow\left\{ P\land\left(\lnot P\leftarrow Q\right)\right\} \lor\left\{ \lnot P\land\left(\lnot P\nleftarrow Q\right)\right\} \\ & \Leftrightarrow\left\{ P\land\left(\lnot P\lor\lnot Q\right)\right\} \lor\left(\lnot P\land P\land Q\right)\\ & \Leftrightarrow P\land\left(\lnot P\lor\lnot Q\right)\\ & \Leftrightarrow P\land\lnot Q\\ & \Leftrightarrow P\nrightarrow Q \end{align*}

(200)

\begin{align*} P\nleftrightarrow\left(\lnot P\nleftrightarrow Q\right) & \Leftrightarrow\left(P\nleftrightarrow\lnot P\right)\nleftrightarrow Q\\ & \Leftrightarrow\left(P\leftrightarrow P\right)\nleftrightarrow Q\\ & \Leftrightarrow1\nleftrightarrow Q\\ & \Leftrightarrow\lnot Q \end{align*}

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