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Theorem List for Metamath Proof Explorer - 101-200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcom15 101 Commutation of antecedents. Swap 1st and 5th. (Contributed by Jeff Hankins, 28-Jun-2009.) (Proof shortened by Wolf Lammen, 29-Jul-2012.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       (𝜏 → (𝜓 → (𝜒 → (𝜃 → (𝜑𝜂)))))

Theoremcom52l 102 Commutation of antecedents. Rotate left twice. (Contributed by Jeff Hankins, 28-Jun-2009.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       (𝜒 → (𝜃 → (𝜏 → (𝜑 → (𝜓𝜂)))))

Theoremcom52r 103 Commutation of antecedents. Rotate right twice. (Contributed by Jeff Hankins, 28-Jun-2009.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       (𝜃 → (𝜏 → (𝜑 → (𝜓 → (𝜒𝜂)))))

Theoremcom5r 104 Commutation of antecedents. Rotate right. (Contributed by Wolf Lammen, 29-Jul-2012.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       (𝜏 → (𝜑 → (𝜓 → (𝜒 → (𝜃𝜂)))))

Theoremimim12 105 Closed form of imim12i 62 and of 3syl 18. (Contributed by BJ, 16-Jul-2019.)
((𝜑𝜓) → ((𝜒𝜃) → ((𝜓𝜒) → (𝜑𝜃))))

Theoremjarr 106 Elimination of a nested antecedent. Sometimes called "Syll-Simp" since it is a syllogism applied to ax-1 6 ("Simplification"). (Contributed by Wolf Lammen, 9-May-2013.)
(((𝜑𝜓) → 𝜒) → (𝜓𝜒))

Theoremjarri 107 Inference associated with jarr 106. Partial converse of ja 189 (the other partial converse being jarli 126). (Contributed by Wolf Lammen, 20-Sep-2013.)
((𝜑𝜓) → 𝜒)       (𝜓𝜒)

Theorempm2.86d 108 Deduction associated with pm2.86 109. (Contributed by NM, 29-Jun-1995.) (Proof shortened by Wolf Lammen, 3-Apr-2013.)
(𝜑 → ((𝜓𝜒) → (𝜓𝜃)))       (𝜑 → (𝜓 → (𝜒𝜃)))

Theorempm2.86 109 Converse of Axiom ax-2 7. Theorem *2.86 of [WhiteheadRussell] p. 108. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 3-Apr-2013.)
(((𝜑𝜓) → (𝜑𝜒)) → (𝜑 → (𝜓𝜒)))

Theorempm2.86i 110 Inference associated with pm2.86 109. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Apr-2013.)
((𝜑𝜓) → (𝜑𝜒))       (𝜑 → (𝜓𝜒))

Theoremloolin 111 The Linearity Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz. See loowoz 112 for an alternate axiom. (Contributed by Mel L. O'Cat, 12-Aug-2004.)
(((𝜑𝜓) → (𝜓𝜑)) → (𝜓𝜑))

Theoremloowoz 112 An alternate for the Linearity Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz loolin 111, due to Barbara Wozniakowska, Reports on Mathematical Logic 10, 129-137 (1978). (Contributed by Mel L. O'Cat, 8-Aug-2004.)
(((𝜑𝜓) → (𝜑𝜒)) → ((𝜓𝜑) → (𝜓𝜒)))

1.2.4  Logical negation

This section makes our first use of the third axiom of propositional calculus, ax-3 8. It introduces logical negation.

Theoremcon4 113 Alias for ax-3 8 to be used instead of it for labeling consistency. Its associated inference is con4i 114 and its associated deduction is con4d 115. (Contributed by BJ, 24-Dec-2020.)
((¬ 𝜑 → ¬ 𝜓) → (𝜓𝜑))

Theoremcon4i 114 Inference associated with con4 113. Its associated inference is mt4 116.

Remark: this can also be proved using notnot 144 followed by nsyl2 143, giving a shorter proof but depending on more axioms (namely, ax-1 6 and ax-2 7). (Contributed by NM, 29-Dec-1992.)

𝜑 → ¬ 𝜓)       (𝜓𝜑)

Theoremcon4d 115 Deduction associated with con4 113. (Contributed by NM, 26-Mar-1995.)
(𝜑 → (¬ 𝜓 → ¬ 𝜒))       (𝜑 → (𝜒𝜓))

Theoremmt4 116 The rule of modus tollens. Inference associated with con4i 114. (Contributed by Wolf Lammen, 12-May-2013.)
𝜑    &   𝜓 → ¬ 𝜑)       𝜓

Theoremmt4d 117 Modus tollens deduction. Deduction form of mt4 116. (Contributed by NM, 9-Jun-2006.)
(𝜑𝜓)    &   (𝜑 → (¬ 𝜒 → ¬ 𝜓))       (𝜑𝜒)

Theoremmt4i 118 Modus tollens inference. (Contributed by Wolf Lammen, 12-May-2013.)
𝜒    &   (𝜑 → (¬ 𝜓 → ¬ 𝜒))       (𝜑𝜓)

Theorempm2.21i 119 A contradiction implies anything. Inference associated with pm2.21 123. Its associated inference is pm2.24ii 120. (Contributed by NM, 16-Sep-1993.)
¬ 𝜑       (𝜑𝜓)

Theorempm2.24ii 120 A contradiction implies anything. Inference associated with pm2.21i 119 and pm2.24i 153. (Contributed by NM, 27-Feb-2008.)
𝜑    &    ¬ 𝜑       𝜓

Theorempm2.21d 121 A contradiction implies anything. Deduction associated with pm2.21 123. (Contributed by NM, 10-Feb-1996.)
(𝜑 → ¬ 𝜓)       (𝜑 → (𝜓𝜒))

Theorempm2.21ddALT 122 Alternate proof of pm2.21dd 198. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   (𝜑 → ¬ 𝜓)       (𝜑𝜒)

Theorempm2.21 123 From a wff and its negation, anything follows. Theorem *2.21 of [WhiteheadRussell] p. 104. Also called the Duns Scotus law. Its commuted form is pm2.24 124 and its associated inference is pm2.21i 119. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 14-Sep-2012.)
𝜑 → (𝜑𝜓))

Theorempm2.24 124 Theorem *2.24 of [WhiteheadRussell] p. 104. Its associated inference is pm2.24i 153. Commuted form of pm2.21 123. (Contributed by NM, 3-Jan-2005.)
(𝜑 → (¬ 𝜑𝜓))

Theoremjarl 125 Elimination of a nested antecedent. (Contributed by Wolf Lammen, 10-May-2013.)
(((𝜑𝜓) → 𝜒) → (¬ 𝜑𝜒))

Theoremjarli 126 Inference associated with jarl 125. Partial converse of ja 189 (the other partial converse being jarri 107). (Contributed by Wolf Lammen, 4-Oct-2013.)
((𝜑𝜓) → 𝜒)       𝜑𝜒)

Theorempm2.18d 127 Deduction form of the Clavius law pm2.18 128. (Contributed by FL, 12-Jul-2009.) (Proof shortened by Andrew Salmon, 7-May-2011.) Revised to shorten pm2.18 128. (Revised by Wolf Lammen, 17-Nov-2023.)
(𝜑 → (¬ 𝜓𝜓))       (𝜑𝜓)

Theorempm2.18 128 Clavius law, or "consequentia mirabilis" ("admirable consequence"). If a formula is implied by its negation, then it is true. Can be used in proofs by contradiction. Theorem *2.18 of [WhiteheadRussell] p. 103. See also the weak Clavius law pm2.01 192. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 17-Nov-2023.)
((¬ 𝜑𝜑) → 𝜑)

Theorempm2.18OLD 129 Obsolete version of pm2.18 128 as of 17-Nov-2023. (Contributed by NM, 29-Dec-1992.) (Proof modification is discouraged.) (New usage is discouraged.)
((¬ 𝜑𝜑) → 𝜑)

Theorempm2.18dOLD 130 Obsolete version of pm2.18d 127 as of 17-Nov-2023. (Contributed by FL, 12-Jul-2009.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (¬ 𝜓𝜓))       (𝜑𝜓)

Theorempm2.18i 131 Inference associated with the Clavius law pm2.18 128. (Contributed by BJ, 30-Mar-2020.)
𝜑𝜑)       𝜑

Theoremnotnotr 132 Double negation elimination. Converse of notnot 144 and one implication of notnotb 318. Theorem *2.14 of [WhiteheadRussell] p. 102. This was the fifth axiom of Frege, specifically Proposition 31 of [Frege1879] p. 44. In classical logic (our logic) this is always true. In intuitionistic logic this is not always true, and formulas for which it is true are called "stable". (Contributed by NM, 29-Dec-1992.) (Proof shortened by David Harvey, 5-Sep-1999.) (Proof shortened by Josh Purinton, 29-Dec-2000.)
(¬ ¬ 𝜑𝜑)

Theoremnotnotri 133 Inference associated with notnotr 132. For a shorter proof using ax-2 7, see notnotriALT 134. (Contributed by NM, 27-Feb-2008.) (Proof shortened by Wolf Lammen, 15-Jul-2021.) Remove dependency on ax-2 7. (Revised by Steven Nguyen, 27-Dec-2022.)
¬ ¬ 𝜑       𝜑

TheoremnotnotriALT 134 Alternate proof of notnotri 133. The proof via notnotr 132 and ax-mp 5 also has three essential steps, but has a total number of steps equal to 8, instead of the present 7, because it has to construct the formula 𝜑 twice and the formula ¬ ¬ 𝜑 once, whereas the present proof has to construct the formula 𝜑 twice and the formula ¬ 𝜑 once, and therefore makes only one use of wn 3 instead of two. This can be checked by running the Metamath command "MM> SHOW PROOF notnotri / NORMAL". (Contributed by NM, 27-Feb-2008.) (Proof shortened by Wolf Lammen, 15-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ ¬ 𝜑       𝜑

Theoremnotnotrd 135 Deduction associated with notnotr 132 and notnotri 133. Double negation elimination rule. A translation of the natural deduction rule ¬ ¬ C , Γ¬ ¬ 𝜓 ⇒ Γ𝜓; see natded 28302. This is Definition NNC in [Pfenning] p. 17. This rule is valid in classical logic (our logic), but not in intuitionistic logic. (Contributed by DAW, 8-Feb-2017.)
(𝜑 → ¬ ¬ 𝜓)       (𝜑𝜓)

Theoremcon2d 136 A contraposition deduction. (Contributed by NM, 19-Aug-1993.)
(𝜑 → (𝜓 → ¬ 𝜒))       (𝜑 → (𝜒 → ¬ 𝜓))

Theoremcon2 137 Contraposition. Theorem *2.03 of [WhiteheadRussell] p. 100. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 12-Feb-2013.)
((𝜑 → ¬ 𝜓) → (𝜓 → ¬ 𝜑))

Theoremmt2d 138 Modus tollens deduction. (Contributed by NM, 4-Jul-1994.)
(𝜑𝜒)    &   (𝜑 → (𝜓 → ¬ 𝜒))       (𝜑 → ¬ 𝜓)

Theoremmt2i 139 Modus tollens inference. (Contributed by NM, 26-Mar-1995.) (Proof shortened by Wolf Lammen, 15-Sep-2012.)
𝜒    &   (𝜑 → (𝜓 → ¬ 𝜒))       (𝜑 → ¬ 𝜓)

Theoremnsyl3 140 A negated syllogism inference. (Contributed by NM, 1-Dec-1995.)
(𝜑 → ¬ 𝜓)    &   (𝜒𝜓)       (𝜒 → ¬ 𝜑)

Theoremcon2i 141 A contraposition inference. Its associated inference is mt2 203. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Mel L. O'Cat, 28-Nov-2008.) (Proof shortened by Wolf Lammen, 13-Jun-2013.)
(𝜑 → ¬ 𝜓)       (𝜓 → ¬ 𝜑)

Theoremnsyl 142 A negated syllogism inference. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Wolf Lammen, 2-Mar-2013.)
(𝜑 → ¬ 𝜓)    &   (𝜒𝜓)       (𝜑 → ¬ 𝜒)

Theoremnsyl2 143 A negated syllogism inference. (Contributed by NM, 26-Jun-1994.) (Proof shortened by Wolf Lammen, 14-Nov-2023.)
(𝜑 → ¬ 𝜓)    &   𝜒𝜓)       (𝜑𝜒)

Theoremnotnot 144 Double negation introduction. Converse of notnotr 132 and one implication of notnotb 318. Theorem *2.12 of [WhiteheadRussell] p. 101. This was the sixth axiom of Frege, specifically Proposition 41 of [Frege1879] p. 47. (Contributed by NM, 28-Dec-1992.) (Proof shortened by Wolf Lammen, 2-Mar-2013.)
(𝜑 → ¬ ¬ 𝜑)

Theoremnotnoti 145 Inference associated with notnot 144. (Contributed by NM, 27-Feb-2008.)
𝜑        ¬ ¬ 𝜑

Theoremnotnotd 146 Deduction associated with notnot 144 and notnoti 145. (Contributed by Jarvin Udandy, 2-Sep-2016.) Avoid biconditional. (Revised by Wolf Lammen, 27-Mar-2021.)
(𝜑𝜓)       (𝜑 → ¬ ¬ 𝜓)

Theoremcon1d 147 A contraposition deduction. (Contributed by NM, 27-Dec-1992.)
(𝜑 → (¬ 𝜓𝜒))       (𝜑 → (¬ 𝜒𝜓))

Theoremcon1 148 Contraposition. Theorem *2.15 of [WhiteheadRussell] p. 102. Its associated inference is con1i 149. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 12-Feb-2013.)
((¬ 𝜑𝜓) → (¬ 𝜓𝜑))

Theoremcon1i 149 A contraposition inference. Inference associated with con1 148. Its associated inference is mt3 204. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Mel L. O'Cat, 28-Nov-2008.) (Proof shortened by Wolf Lammen, 19-Jun-2013.)
𝜑𝜓)       𝜓𝜑)

Theoremmt3d 150 Modus tollens deduction. (Contributed by NM, 26-Mar-1995.)
(𝜑 → ¬ 𝜒)    &   (𝜑 → (¬ 𝜓𝜒))       (𝜑𝜓)

Theoremmt3i 151 Modus tollens inference. (Contributed by NM, 26-Mar-1995.) (Proof shortened by Wolf Lammen, 15-Sep-2012.)
¬ 𝜒    &   (𝜑 → (¬ 𝜓𝜒))       (𝜑𝜓)

Theoremnsyl2OLD 152 Obsolete version of nsyl2 143 as of 14-Nov-2023. (Contributed by NM, 26-Jun-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ¬ 𝜓)    &   𝜒𝜓)       (𝜑𝜒)

Theorempm2.24i 153 Inference associated with pm2.24 124. Its associated inference is pm2.24ii 120. (Contributed by NM, 20-Aug-2001.)
𝜑       𝜑𝜓)

Theorempm2.24d 154 Deduction form of pm2.24 124. (Contributed by NM, 30-Jan-2006.)
(𝜑𝜓)       (𝜑 → (¬ 𝜓𝜒))

Theoremcon3d 155 A contraposition deduction. Deduction form of con3 156. (Contributed by NM, 10-Jan-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → (¬ 𝜒 → ¬ 𝜓))

Theoremcon3 156 Contraposition. Theorem *2.16 of [WhiteheadRussell] p. 103. This was the fourth axiom of Frege, specifically Proposition 28 of [Frege1879] p. 43. Its associated inference is con3i 157. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 13-Feb-2013.)
((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))

Theoremcon3i 157 A contraposition inference. Inference associated with con3 156. Its associated inference is mto 200. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 20-Jun-2013.)
(𝜑𝜓)       𝜓 → ¬ 𝜑)

Theoremcon3rr3 158 Rotate through consequent right. (Contributed by Wolf Lammen, 3-Nov-2013.)
(𝜑 → (𝜓𝜒))       𝜒 → (𝜑 → ¬ 𝜓))

Theoremnsyld 159 A negated syllogism deduction. (Contributed by NM, 9-Apr-2005.)
(𝜑 → (𝜓 → ¬ 𝜒))    &   (𝜑 → (𝜏𝜒))       (𝜑 → (𝜓 → ¬ 𝜏))

Theoremnsyli 160 A negated syllogism inference. (Contributed by NM, 3-May-1994.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → ¬ 𝜒)       (𝜑 → (𝜃 → ¬ 𝜓))

Theoremnsyl4 161 A negated syllogism inference. (Contributed by NM, 15-Feb-1996.)
(𝜑𝜓)    &   𝜑𝜒)       𝜒𝜓)

Theoremnsyl5 162 A negated syllogism inference. (Contributed by Wolf Lammen, 20-May-2024.)
(𝜑𝜓)    &   𝜑𝜒)       𝜓𝜒)

Theorempm3.2im 163 Theorem *3.2 of [WhiteheadRussell] p. 111, expressed with primitive connectives (see pm3.2 473). (Contributed by NM, 29-Dec-1992.) (Proof shortened by Josh Purinton, 29-Dec-2000.)
(𝜑 → (𝜓 → ¬ (𝜑 → ¬ 𝜓)))

Theoremjc 164 Deduction joining the consequents of two premises. A deduction associated with pm3.2im 163. (Contributed by NM, 28-Dec-1992.)
(𝜑𝜓)    &   (𝜑𝜒)       (𝜑 → ¬ (𝜓 → ¬ 𝜒))

Theoremjcn 165 Theorem joining the consequents of two premises. Theorem 8 of [Margaris] p. 60. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.)
(𝜑 → (¬ 𝜓 → ¬ (𝜑𝜓)))

Theoremjcnd 166 Deduction joining the consequents of two premises. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Proof shortened by Wolf Lammen, 10-Apr-2024.)
(𝜑𝜓)    &   (𝜑 → ¬ 𝜒)       (𝜑 → ¬ (𝜓𝜒))

Theoremimpi 167 An importation inference. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 20-Jul-2013.)
(𝜑 → (𝜓𝜒))       (¬ (𝜑 → ¬ 𝜓) → 𝜒)

Theoremexpi 168 An exportation inference. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Mel L. O'Cat, 28-Nov-2008.)
(¬ (𝜑 → ¬ 𝜓) → 𝜒)       (𝜑 → (𝜓𝜒))

Theoremsimprim 169 Simplification. Similar to Theorem *3.27 (Simp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 13-Nov-2012.)
(¬ (𝜑 → ¬ 𝜓) → 𝜓)

Theoremsimplim 170 Simplification. Similar to Theorem *3.26 (Simp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 21-Jul-2012.)
(¬ (𝜑𝜓) → 𝜑)

Theorempm2.5g 171 General instance of Theorem *2.5 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 9-Oct-2012.)
(¬ (𝜑𝜓) → (¬ 𝜑𝜒))

Theorempm2.5 172 Theorem *2.5 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
(¬ (𝜑𝜓) → (¬ 𝜑𝜓))

Theoremconax1 173 Contrapositive of ax-1 6. (Contributed by BJ, 28-Oct-2023.)
(¬ (𝜑𝜓) → ¬ 𝜓)

Theoremconax1k 174 Weakening of conax1 173. General instance of pm2.51 175 and of pm2.52 176. (Contributed by BJ, 28-Oct-2023.)
(¬ (𝜑𝜓) → (𝜒 → ¬ 𝜓))

Theorempm2.51 175 Theorem *2.51 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
(¬ (𝜑𝜓) → (𝜑 → ¬ 𝜓))

Theorempm2.52 176 Theorem *2.52 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 8-Oct-2012.)
(¬ (𝜑𝜓) → (¬ 𝜑 → ¬ 𝜓))

Theorempm2.521g 177 A general instance of Theorem *2.521 of [WhiteheadRussell] p. 107. (Contributed by BJ, 28-Oct-2023.)
(¬ (𝜑𝜓) → (𝜓𝜒))

Theorempm2.521g2 178 A general instance of Theorem *2.521 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 8-Oct-2012.)
(¬ (𝜑𝜓) → (𝜒𝜑))

Theorempm2.521 179 Theorem *2.521 of [WhiteheadRussell] p. 107. Instance of pm2.521g 177 and of pm2.521g2 178. (Contributed by NM, 3-Jan-2005.)
(¬ (𝜑𝜓) → (𝜓𝜑))

Theoremexpt 180 Exportation theorem pm3.3 452 (closed form of ex 416) expressed with primitive connectives. (Contributed by NM, 28-Dec-1992.)
((¬ (𝜑 → ¬ 𝜓) → 𝜒) → (𝜑 → (𝜓𝜒)))

Theoremimpt 181 Importation theorem pm3.1 989 (closed form of imp 410) expressed with primitive connectives. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 20-Jul-2013.)
((𝜑 → (𝜓𝜒)) → (¬ (𝜑 → ¬ 𝜓) → 𝜒))

Theorempm2.61d 182 Deduction eliminating an antecedent. (Contributed by NM, 27-Apr-1994.) (Proof shortened by Wolf Lammen, 12-Sep-2013.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (¬ 𝜓𝜒))       (𝜑𝜒)

Theorempm2.61d1 183 Inference eliminating an antecedent. (Contributed by NM, 15-Jul-2005.)
(𝜑 → (𝜓𝜒))    &   𝜓𝜒)       (𝜑𝜒)

Theorempm2.61d2 184 Inference eliminating an antecedent. (Contributed by NM, 18-Aug-1993.)
(𝜑 → (¬ 𝜓𝜒))    &   (𝜓𝜒)       (𝜑𝜒)

Theorempm2.61i 185 Inference eliminating an antecedent. (Contributed by NM, 5-Apr-1994.) (Proof shortened by Wolf Lammen, 19-Nov-2023.)
(𝜑𝜓)    &   𝜑𝜓)       𝜓

Theorempm2.61ii 186 Inference eliminating two antecedents. (Contributed by NM, 4-Jan-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.)
𝜑 → (¬ 𝜓𝜒))    &   (𝜑𝜒)    &   (𝜓𝜒)       𝜒

Theorempm2.61nii 187 Inference eliminating two antecedents. (Contributed by NM, 13-Jul-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 13-Nov-2012.)
(𝜑 → (𝜓𝜒))    &   𝜑𝜒)    &   𝜓𝜒)       𝜒

Theorempm2.61iii 188 Inference eliminating three antecedents. (Contributed by NM, 2-Jan-2002.) (Proof shortened by Wolf Lammen, 22-Sep-2013.)
𝜑 → (¬ 𝜓 → (¬ 𝜒𝜃)))    &   (𝜑𝜃)    &   (𝜓𝜃)    &   (𝜒𝜃)       𝜃

Theoremja 189 Inference joining the antecedents of two premises. For partial converses, see jarri 107 and jarli 126. (Contributed by NM, 24-Jan-1993.) (Proof shortened by Mel L. O'Cat, 19-Feb-2008.)
𝜑𝜒)    &   (𝜓𝜒)       ((𝜑𝜓) → 𝜒)

Theoremjad 190 Deduction form of ja 189. (Contributed by Scott Fenton, 13-Dec-2010.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
(𝜑 → (¬ 𝜓𝜃))    &   (𝜑 → (𝜒𝜃))       (𝜑 → ((𝜓𝜒) → 𝜃))

Theorempm2.61iOLD 191 Obsolete version of pm2.61i 185 as of 19-Nov-2023. (Contributed by NM, 5-Apr-1994.) (Proof shortened by Wolf Lammen, 12-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   𝜑𝜓)       𝜓

Theorempm2.01 192 Weak Clavius law. If a formula implies its negation, then it is false. A form of "reductio ad absurdum", which can be used in proofs by contradiction. Theorem *2.01 of [WhiteheadRussell] p. 100. Provable in minimal calculus, contrary to the Clavius law pm2.18 128. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Mel L. O'Cat, 21-Nov-2008.) (Proof shortened by Wolf Lammen, 31-Oct-2012.)
((𝜑 → ¬ 𝜑) → ¬ 𝜑)

Theorempm2.01d 193 Deduction based on reductio ad absurdum. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 5-Mar-2013.)
(𝜑 → (𝜓 → ¬ 𝜓))       (𝜑 → ¬ 𝜓)

Theorempm2.6 194 Theorem *2.6 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
((¬ 𝜑𝜓) → ((𝜑𝜓) → 𝜓))

Theorempm2.61 195 Theorem *2.61 of [WhiteheadRussell] p. 107. Useful for eliminating an antecedent. (Contributed by NM, 4-Jan-1993.) (Proof shortened by Wolf Lammen, 22-Sep-2013.)
((𝜑𝜓) → ((¬ 𝜑𝜓) → 𝜓))

Theorempm2.65 196 Theorem *2.65 of [WhiteheadRussell] p. 107. Proof by contradiction. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Wolf Lammen, 8-Mar-2013.)
((𝜑𝜓) → ((𝜑 → ¬ 𝜓) → ¬ 𝜑))

Theorempm2.65i 197 Inference for proof by contradiction. (Contributed by NM, 18-May-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)
(𝜑𝜓)    &   (𝜑 → ¬ 𝜓)        ¬ 𝜑

Theorempm2.21dd 198 A contradiction implies anything. Deduction from pm2.21 123. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof shortened by Wolf Lammen, 22-Jul-2019.)
(𝜑𝜓)    &   (𝜑 → ¬ 𝜓)       (𝜑𝜒)

Theorempm2.65d 199 Deduction for proof by contradiction. (Contributed by NM, 26-Jun-1994.) (Proof shortened by Wolf Lammen, 26-May-2013.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓 → ¬ 𝜒))       (𝜑 → ¬ 𝜓)

Theoremmto 200 The rule of modus tollens. The rule says, "if 𝜓 is not true, and 𝜑 implies 𝜓, then 𝜑 must also be not true". Modus tollens is short for "modus tollendo tollens", a Latin phrase that means "the mode that by denying denies" - remark in [Sanford] p. 39. It is also called denying the consequent. Modus tollens is closely related to modus ponens ax-mp 5. Note that this rule is also valid in intuitionistic logic. Inference associated with con3i 157. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)
¬ 𝜓    &   (𝜑𝜓)        ¬ 𝜑

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