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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 as a partial converse of ja 173 (the other being jarl 175). (Contributed by Wolf Lammen, 9-May-2013.)
(((𝜑𝜓) → 𝜒) → (𝜓𝜒))
 
Theorempm2.86d 107 Deduction associated with pm2.86 108. (Contributed by NM, 29-Jun-1995.) (Proof shortened by Wolf Lammen, 3-Apr-2013.)
(𝜑 → ((𝜓𝜒) → (𝜓𝜃)))       (𝜑 → (𝜓 → (𝜒𝜃)))
 
Theorempm2.86 108 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 109 Inference associated with pm2.86 108. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Apr-2013.)
((𝜑𝜓) → (𝜑𝜒))       (𝜑 → (𝜓𝜒))
 
Theoremloolin 110 The Linearity Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz. See loowoz 111 for an alternate axiom. (Contributed by Mel L. O'Cat, 12-Aug-2004.)
(((𝜑𝜓) → (𝜓𝜑)) → (𝜓𝜑))
 
Theoremloowoz 111 An alternate for the Linearity Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz loolin 110, 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.

 
Theoremcon4 112 Alias for ax-3 8 to be used instead of it for labeling consistency. Its associated inference is con4i 113 and its associated deduction is con4d 114. (Contributed by BJ, 24-Dec-2020.)
((¬ 𝜑 → ¬ 𝜓) → (𝜓𝜑))
 
Theoremcon4i 113 Inference associated with con4 112. Its associated inference is mt4 115.

Remark: this can also be proved using notnot 136 followed by nsyl2 142, 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 114 Deduction associated with con4 112. (Contributed by NM, 26-Mar-1995.)
(𝜑 → (¬ 𝜓 → ¬ 𝜒))       (𝜑 → (𝜒𝜓))
 
Theoremmt4 115 The rule of modus tollens. Inference associated with con4i 113. (Contributed by Wolf Lammen, 12-May-2013.)
𝜑    &   𝜓 → ¬ 𝜑)       𝜓
 
Theorempm2.21i 116 A contradiction implies anything. Inference associated with pm2.21 120. Its associated inference is pm2.24ii 117. (Contributed by NM, 16-Sep-1993.)
¬ 𝜑       (𝜑𝜓)
 
Theorempm2.24ii 117 A contradiction implies anything. Inference associated with pm2.21i 116 and pm2.24i 146. (Contributed by NM, 27-Feb-2008.)
𝜑    &    ¬ 𝜑       𝜓
 
Theorempm2.21d 118 A contradiction implies anything. Deduction associated with pm2.21 120. (Contributed by NM, 10-Feb-1996.)
(𝜑 → ¬ 𝜓)       (𝜑 → (𝜓𝜒))
 
Theorempm2.21ddALT 119 Alternate proof of pm2.21dd 186. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   (𝜑 → ¬ 𝜓)       (𝜑𝜒)
 
Theorempm2.21 120 From a wff and its negation, anything is true. Theorem *2.21 of [WhiteheadRussell] p. 104. Also called the Duns Scotus law. Its associated inference is pm2.21i 116. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 14-Sep-2012.)
𝜑 → (𝜑𝜓))
 
Theorempm2.24 121 Theorem *2.24 of [WhiteheadRussell] p. 104. Its associated inference is pm2.24i 146. (Contributed by NM, 3-Jan-2005.)
(𝜑 → (¬ 𝜑𝜓))
 
Theorempm2.18 122 Proof by contradiction. Theorem *2.18 of [WhiteheadRussell] p. 103. Also called the Law of Clavius. See also pm2.01 180. (Contributed by NM, 29-Dec-1992.)
((¬ 𝜑𝜑) → 𝜑)
 
Theorempm2.18i 123 Inference associated with pm2.18 122. (Contributed by BJ, 30-Mar-2020.)
𝜑𝜑)       𝜑
 
Theorempm2.18d 124 Deduction based on reductio ad absurdum. (Contributed by FL, 12-Jul-2009.) (Proof shortened by Andrew Salmon, 7-May-2011.)
(𝜑 → (¬ 𝜓𝜓))       (𝜑𝜓)
 
Theoremnotnotr 125 Double negation elimination. Converse of notnot 136 and one implication of notnotb 304. 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 126 Inference associated with notnotr 125.

Remark: the proof via notnotr 125 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, whereas the present proof has to construct the formula 𝜑 twice and the formula ¬ 𝜑, and therefore makes only one use of wn 3 instead of two. This can be checked by running the Metamath command "SHOW PROOF notnotri / NORMAL". (Contributed by NM, 27-Feb-2008.) (Proof shortened by Wolf Lammen, 15-Jul-2021.)

¬ ¬ 𝜑       𝜑
 
TheoremnotnotriOLD 127 Obsolete proof of notnotri 126 as of 15-Jul-2021 . (Contributed by NM, 27-Feb-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ ¬ 𝜑       𝜑
 
Theoremnotnotrd 128 Deduction associated with notnotr 125 and notnotri 126. Double negation elimination rule. A translation of the natural deduction rule ¬ ¬ C , Γ¬ ¬ 𝜓 ⇒ Γ𝜓; see natded 27567. 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 129 A contraposition deduction. (Contributed by NM, 19-Aug-1993.)
(𝜑 → (𝜓 → ¬ 𝜒))       (𝜑 → (𝜒 → ¬ 𝜓))
 
Theoremcon2 130 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 131 Modus tollens deduction. (Contributed by NM, 4-Jul-1994.)
(𝜑𝜒)    &   (𝜑 → (𝜓 → ¬ 𝜒))       (𝜑 → ¬ 𝜓)
 
Theoremmt2i 132 Modus tollens inference. (Contributed by NM, 26-Mar-1995.) (Proof shortened by Wolf Lammen, 15-Sep-2012.)
𝜒    &   (𝜑 → (𝜓 → ¬ 𝜒))       (𝜑 → ¬ 𝜓)
 
Theoremnsyl3 133 A negated syllogism inference. (Contributed by NM, 1-Dec-1995.)
(𝜑 → ¬ 𝜓)    &   (𝜒𝜓)       (𝜒 → ¬ 𝜑)
 
Theoremcon2i 134 A contraposition inference. Its associated inference is mt2 191. (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 135 A negated syllogism inference. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Wolf Lammen, 2-Mar-2013.)
(𝜑 → ¬ 𝜓)    &   (𝜒𝜓)       (𝜑 → ¬ 𝜒)
 
Theoremnotnot 136 Double negation introduction. Converse of notnotr 125 and one implication of notnotb 304. 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 137 Inference associated with notnot 136. (Contributed by NM, 27-Feb-2008.)
𝜑        ¬ ¬ 𝜑
 
Theoremnotnotd 138 Deduction associated with notnot 136 and notnoti 137. (Contributed by Jarvin Udandy, 2-Sep-2016.) Avoid biconditional. (Revised by Wolf Lammen, 27-Mar-2021.)
(𝜑𝜓)       (𝜑 → ¬ ¬ 𝜓)
 
Theoremcon1d 139 A contraposition deduction. (Contributed by NM, 27-Dec-1992.)
(𝜑 → (¬ 𝜓𝜒))       (𝜑 → (¬ 𝜒𝜓))
 
Theoremmt3d 140 Modus tollens deduction. (Contributed by NM, 26-Mar-1995.)
(𝜑 → ¬ 𝜒)    &   (𝜑 → (¬ 𝜓𝜒))       (𝜑𝜓)
 
Theoremmt3i 141 Modus tollens inference. (Contributed by NM, 26-Mar-1995.) (Proof shortened by Wolf Lammen, 15-Sep-2012.)
¬ 𝜒    &   (𝜑 → (¬ 𝜓𝜒))       (𝜑𝜓)
 
Theoremnsyl2 142 A negated syllogism inference. (Contributed by NM, 26-Jun-1994.)
(𝜑 → ¬ 𝜓)    &   𝜒𝜓)       (𝜑𝜒)
 
Theoremcon1 143 Contraposition. Theorem *2.15 of [WhiteheadRussell] p. 102. Its associated inference is con1i 144. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 12-Feb-2013.)
((¬ 𝜑𝜓) → (¬ 𝜓𝜑))
 
Theoremcon1i 144 A contraposition inference. Inference associated with con1 143. Its associated inference is mt3 192. (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)
 
Theoremcon4iOLD 145 Obsolete proof of con4i 113 as of 15-Jul-2021. This shorter proof has been reverted to its original to avoid a dependency on ax-1 6 and ax-2 7. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 21-Jun-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑 → ¬ 𝜓)       (𝜓𝜑)
 
Theorempm2.24i 146 Inference associated with pm2.24 121. Its associated inference is pm2.24ii 117. (Contributed by NM, 20-Aug-2001.)
𝜑       𝜑𝜓)
 
Theorempm2.24d 147 Deduction form of pm2.24 121. (Contributed by NM, 30-Jan-2006.)
(𝜑𝜓)       (𝜑 → (¬ 𝜓𝜒))
 
Theoremcon3d 148 A contraposition deduction. Deduction form of con3 149. (Contributed by NM, 10-Jan-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → (¬ 𝜒 → ¬ 𝜓))
 
Theoremcon3 149 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 150. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 13-Feb-2013.)
((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))
 
Theoremcon3i 150 A contraposition inference. Inference associated with con3 149. Its associated inference is mto 188. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 20-Jun-2013.)
(𝜑𝜓)       𝜓 → ¬ 𝜑)
 
Theoremcon3rr3 151 Rotate through consequent right. (Contributed by Wolf Lammen, 3-Nov-2013.)
(𝜑 → (𝜓𝜒))       𝜒 → (𝜑 → ¬ 𝜓))
 
Theoremmt4d 152 Modus tollens deduction. Deduction form of mt4 115. (Contributed by NM, 9-Jun-2006.)
(𝜑𝜓)    &   (𝜑 → (¬ 𝜒 → ¬ 𝜓))       (𝜑𝜒)
 
Theoremmt4i 153 Modus tollens inference. (Contributed by Wolf Lammen, 12-May-2013.)
𝜒    &   (𝜑 → (¬ 𝜓 → ¬ 𝜒))       (𝜑𝜓)
 
Theoremnsyld 154 A negated syllogism deduction. (Contributed by NM, 9-Apr-2005.)
(𝜑 → (𝜓 → ¬ 𝜒))    &   (𝜑 → (𝜏𝜒))       (𝜑 → (𝜓 → ¬ 𝜏))
 
Theoremnsyli 155 A negated syllogism inference. (Contributed by NM, 3-May-1994.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → ¬ 𝜒)       (𝜑 → (𝜃 → ¬ 𝜓))
 
Theoremnsyl4 156 A negated syllogism inference. (Contributed by NM, 15-Feb-1996.)
(𝜑𝜓)    &   𝜑𝜒)       𝜒𝜓)
 
Theorempm3.2im 157 Theorem *3.2 of [WhiteheadRussell] p. 111, expressed with primitive connectives (see pm3.2 462). (Contributed by NM, 29-Dec-1992.) (Proof shortened by Josh Purinton, 29-Dec-2000.)
(𝜑 → (𝜓 → ¬ (𝜑 → ¬ 𝜓)))
 
Theoremmth8 158 Theorem 8 of [Margaris] p. 60. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.)
(𝜑 → (¬ 𝜓 → ¬ (𝜑𝜓)))
 
Theoremjc 159 Deduction joining the consequents of two premises. A deduction associated with pm3.2im 157. (Contributed by NM, 28-Dec-1992.)
(𝜑𝜓)    &   (𝜑𝜒)       (𝜑 → ¬ (𝜓 → ¬ 𝜒))
 
Theoremimpi 160 An importation inference. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 20-Jul-2013.)
(𝜑 → (𝜓𝜒))       (¬ (𝜑 → ¬ 𝜓) → 𝜒)
 
Theoremexpi 161 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 162 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 163 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)
 
Theorempm2.5 164 Theorem *2.5 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 9-Oct-2012.)
(¬ (𝜑𝜓) → (¬ 𝜑𝜓))
 
Theorempm2.51 165 Theorem *2.51 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
(¬ (𝜑𝜓) → (𝜑 → ¬ 𝜓))
 
Theorempm2.521 166 Theorem *2.521 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 8-Oct-2012.)
(¬ (𝜑𝜓) → (𝜓𝜑))
 
Theorempm2.52 167 Theorem *2.52 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 8-Oct-2012.)
(¬ (𝜑𝜓) → (¬ 𝜑 → ¬ 𝜓))
 
Theoremexpt 168 Exportation theorem ex 449 expressed with primitive connectives. (Contributed by NM, 28-Dec-1992.)
((¬ (𝜑 → ¬ 𝜓) → 𝜒) → (𝜑 → (𝜓𝜒)))
 
Theoremimpt 169 Importation theorem imp 444 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 170 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 171 Inference eliminating an antecedent. (Contributed by NM, 15-Jul-2005.)
(𝜑 → (𝜓𝜒))    &   𝜓𝜒)       (𝜑𝜒)
 
Theorempm2.61d2 172 Inference eliminating an antecedent. (Contributed by NM, 18-Aug-1993.)
(𝜑 → (¬ 𝜓𝜒))    &   (𝜓𝜒)       (𝜑𝜒)
 
Theoremja 173 Inference joining the antecedents of two premises. For partial converses, see jarr 106 and jarl 175. (Contributed by NM, 24-Jan-1993.) (Proof shortened by Mel L. O'Cat, 19-Feb-2008.)
𝜑𝜒)    &   (𝜓𝜒)       ((𝜑𝜓) → 𝜒)
 
Theoremjad 174 Deduction form of ja 173. (Contributed by Scott Fenton, 13-Dec-2010.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
(𝜑 → (¬ 𝜓𝜃))    &   (𝜑 → (𝜒𝜃))       (𝜑 → ((𝜓𝜒) → 𝜃))
 
Theoremjarl 175 Elimination of a nested antecedent as a partial converse of ja 173 (the other being jarr 106). (Contributed by Wolf Lammen, 10-May-2013.)
(((𝜑𝜓) → 𝜒) → (¬ 𝜑𝜒))
 
Theorempm2.61i 176 Inference eliminating an antecedent. (Contributed by NM, 5-Apr-1994.) (Proof shortened by Wolf Lammen, 12-Sep-2013.)
(𝜑𝜓)    &   𝜑𝜓)       𝜓
 
Theorempm2.61ii 177 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 178 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 179 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
 
Theorempm2.01 180 Reductio ad absurdum. Theorem *2.01 of [WhiteheadRussell] p. 100. Also called the weak Clavius law, and provable in minimal calculus, contrary to the Clavius law pm2.18 122. (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 181 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)
 
Theorempm2.6 182 Theorem *2.6 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
((¬ 𝜑𝜓) → ((𝜑𝜓) → 𝜓))
 
Theorempm2.61 183 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))
 
Theorempm2.65 184 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 185 Inference rule for proof by contradiction. (Contributed by NM, 18-May-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)
(𝜑𝜓)    &   (𝜑 → ¬ 𝜓)        ¬ 𝜑
 
Theorempm2.21dd 186 A contradiction implies anything. Deduction from pm2.21 120. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof shortened by Wolf Lammen, 22-Jul-2019.)
(𝜑𝜓)    &   (𝜑 → ¬ 𝜓)       (𝜑𝜒)
 
Theorempm2.65d 187 Deduction rule 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 188 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 150. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)
¬ 𝜓    &   (𝜑𝜓)        ¬ 𝜑
 
Theoremmtod 189 Modus tollens deduction. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)
(𝜑 → ¬ 𝜒)    &   (𝜑 → (𝜓𝜒))       (𝜑 → ¬ 𝜓)
 
Theoremmtoi 190 Modus tollens inference. (Contributed by NM, 5-Jul-1994.) (Proof shortened by Wolf Lammen, 15-Sep-2012.)
¬ 𝜒    &   (𝜑 → (𝜓𝜒))       (𝜑 → ¬ 𝜓)
 
Theoremmt2 191 A rule similar to modus tollens. Inference associated with con2i 134. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 10-Sep-2013.)
𝜓    &   (𝜑 → ¬ 𝜓)        ¬ 𝜑
 
Theoremmt3 192 A rule similar to modus tollens. Inference associated with con1i 144. (Contributed by NM, 18-May-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)
¬ 𝜓    &   𝜑𝜓)       𝜑
 
Theorempeirce 193 Peirce's axiom. This odd-looking theorem is the "difference" between an intuitionistic system of propositional calculus and a classical system and is not accepted by intuitionists. When Peirce's axiom is added to an intuitionistic system, the system becomes equivalent to our classical system ax-1 6 through ax-3 8. A notable fact about this theorem is that it requires ax-3 8 for its proof even though the result has no negation connectives in it. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 9-Oct-2012.)
(((𝜑𝜓) → 𝜑) → 𝜑)
 
Theoremlooinv 194 The Inversion Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz. Using dfor2 426, we can see that this essentially expresses "disjunction commutes." Theorem *2.69 of [WhiteheadRussell] p. 108. It is a special instance of the axiom "Roll", see peirceroll 85. (Contributed by NM, 12-Aug-2004.)
(((𝜑𝜓) → 𝜓) → ((𝜓𝜑) → 𝜑))
 
Theorembijust 195 Theorem used to justify definition of biconditional df-bi 197. (Contributed by NM, 11-May-1999.) (Proof shortened by Josh Purinton, 29-Dec-2000.)
¬ ((¬ ((𝜑𝜓) → ¬ (𝜓𝜑)) → ¬ ((𝜑𝜓) → ¬ (𝜓𝜑))) → ¬ (¬ ((𝜑𝜓) → ¬ (𝜓𝜑)) → ¬ ((𝜑𝜓) → ¬ (𝜓𝜑))))
 
1.2.5  Logical equivalence

The definition df-bi 197 in this section is our first definition, which introduces and defines the biconditional connective . We define a wff of the form (𝜑𝜓) as an abbreviation for ¬ ((𝜑𝜓) → ¬ (𝜓𝜑)).

Unlike most traditional developments, we have chosen not to have a separate symbol such as "Df." to mean "is defined as." Instead, we will later use the biconditional connective for this purpose (df-or 384 is its first use), as it allows us to use logic to manipulate definitions directly. This greatly simplifies many proofs since it eliminates the need for a separate mechanism for introducing and eliminating definitions.

 
Syntaxwb 196 Extend our wff definition to include the biconditional connective.
wff (𝜑𝜓)
 
Definitiondf-bi 197 Define the biconditional (logical 'iff').

The definition df-bi 197 in this section is our first definition, which introduces and defines the biconditional connective . We define a wff of the form (𝜑𝜓) as an abbreviation for ¬ ((𝜑𝜓) → ¬ (𝜓𝜑)).

Unlike most traditional developments, we have chosen not to have a separate symbol such as "Df." to mean "is defined as." Instead, we will later use the biconditional connective for this purpose (df-or 384 is its first use), as it allows us to use logic to manipulate definitions directly. This greatly simplifies many proofs since it eliminates the need for a separate mechanism for introducing and eliminating definitions. Of course, we cannot use this mechanism to define the biconditional itself, since it hasn't been introduced yet. Instead, we use a more general form of definition, described as follows.

In its most general form, a definition is simply an assertion that introduces a new symbol (or a new combination of existing symbols, as in df-3an 1074) that is eliminable and does not strengthen the existing language. The latter requirement means that the set of provable statements not containing the new symbol (or new combination) should remain exactly the same after the definition is introduced. Our definition of the biconditional may look unusual compared to most definitions, but it strictly satisfies these requirements.

The justification for our definition is that if we mechanically replace (𝜑𝜓) (the definiendum i.e. the thing being defined) with ¬ ((𝜑𝜓) → ¬ (𝜓𝜑)) (the definiens i.e. the defining expression) in the definition, the definition becomes the previously proved theorem bijust 195. It is impossible to use df-bi 197 to prove any statement expressed in the original language that can't be proved from the original axioms, because if we simply replace each instance of df-bi 197 in the proof with the corresponding bijust 195 instance, we will end up with a proof from the original axioms.

Note that from Metamath's point of view, a definition is just another axiom - i.e. an assertion we claim to be true - but from our high level point of view, we are not strengthening the language. To indicate this fact, we prefix definition labels with "df-" instead of "ax-". (This prefixing is an informal convention that means nothing to the Metamath proof verifier; it is just a naming convention for human readability.)

After we define the constant true (df-tru 1631) and the constant false (df-fal 1634), we will be able to prove these truth table values: ((⊤ ↔ ⊤) ↔ ⊤) (trubitru 1665), ((⊤ ↔ ⊥) ↔ ⊥) (trubifal 1667), ((⊥ ↔ ⊤) ↔ ⊥) (falbitru 1666), and ((⊥ ↔ ⊥) ↔ ⊤) (falbifal 1668).

See dfbi1 203, dfbi2 663, and dfbi3 1036 for theorems suggesting typical textbook definitions of , showing that our definition has the properties we expect. Theorem dfbi1 203 is particularly useful if we want to eliminate from an expression to convert it to primitives. Theorem dfbi 664 shows this definition rewritten in an abbreviated form after conjunction is introduced, for easier understanding.

Contrast with (df-or 384), (wi 4), (df-nan 1593), and (df-xor 1610) . In some sense returns true if two truth values are equal; = (df-cleq 2749) returns true if two classes are equal. (Contributed by NM, 27-Dec-1992.)

¬ (((𝜑𝜓) → ¬ ((𝜑𝜓) → ¬ (𝜓𝜑))) → ¬ (¬ ((𝜑𝜓) → ¬ (𝜓𝜑)) → (𝜑𝜓)))
 
Theoremimpbi 198 Property of the biconditional connective. (Contributed by NM, 11-May-1999.)
((𝜑𝜓) → ((𝜓𝜑) → (𝜑𝜓)))
 
Theoremimpbii 199 Infer an equivalence from an implication and its converse. Inference associated with impbi 198. (Contributed by NM, 29-Dec-1992.)
(𝜑𝜓)    &   (𝜓𝜑)       (𝜑𝜓)
 
Theoremimpbidd 200 Deduce an equivalence from two implications. Double deduction associated with impbi 198 and impbii 199. Deduction associated with impbid 202. (Contributed by Rodolfo Medina, 12-Oct-2010.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜑 → (𝜓 → (𝜃𝜒)))       (𝜑 → (𝜓 → (𝜒𝜃)))
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