Home Metamath Proof ExplorerTheorem List (p. 2 of 424) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-27159) Hilbert Space Explorer (27160-28684) Users' Mathboxes (28685-42360)

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

Theoremcom52r 101 Commutation of antecedents. Rotate right twice. (Contributed by Jeff Hankins, 28-Jun-2009.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       (𝜃 → (𝜏 → (𝜑 → (𝜓 → (𝜒𝜂)))))

Theoremcom5r 102 Commutation of antecedents. Rotate right. (Contributed by Wolf Lammen, 29-Jul-2012.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       (𝜏 → (𝜑 → (𝜓 → (𝜒 → (𝜃𝜂)))))

Theoremimim12 103 Closed form of imim12i 60 and of 3syl 18. (Contributed by BJ, 16-Jul-2019.)
((𝜑𝜓) → ((𝜒𝜃) → ((𝜓𝜒) → (𝜑𝜃))))

Theoremjarr 104 Elimination of a nested antecedent as a partial converse of ja 172 (the other being jarl 174). (Contributed by Wolf Lammen, 9-May-2013.)
(((𝜑𝜓) → 𝜒) → (𝜓𝜒))

Theorempm2.86d 105 Deduction associated with pm2.86 106. (Contributed by NM, 29-Jun-1995.) (Proof shortened by Wolf Lammen, 3-Apr-2013.)
(𝜑 → ((𝜓𝜒) → (𝜓𝜃)))       (𝜑 → (𝜓 → (𝜒𝜃)))

Theorempm2.86 106 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 107 Inference associated with pm2.86 106. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Apr-2013.)
((𝜑𝜓) → (𝜑𝜒))       (𝜑 → (𝜓𝜒))

Theorempm2.86iALT 108 Alternate proof of pm2.86i 107 with only three essential steps. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 19-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → (𝜑𝜒))       (𝜑 → (𝜓𝜒))

Theoremloolin 109 The Linearity Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz. See loowoz 110 for an alternate axiom. (Contributed by Mel L. O'Cat, 12-Aug-2004.)
(((𝜑𝜓) → (𝜓𝜑)) → (𝜓𝜑))

Theoremloowoz 110 An alternate for the Linearity Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz loolin 109, 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 111 Alias for ax-3 8 to be used instead of it for labeling consistency. Its associated inference is con4i 112 and its associated deduction is con4d 113. (Contributed by BJ, 24-Dec-2020.)
((¬ 𝜑 → ¬ 𝜓) → (𝜓𝜑))

Theoremcon4i 112 Inference associated with con4 111. Its associated inference is mt4 114. (Contributed by NM, 29-Dec-1992.)
𝜑 → ¬ 𝜓)       (𝜓𝜑)

Theoremcon4d 113 Deduction associated with con4 111. (Contributed by NM, 26-Mar-1995.)
(𝜑 → (¬ 𝜓 → ¬ 𝜒))       (𝜑 → (𝜒𝜓))

Theoremmt4 114 The rule of modus tollens. Inference associated with con4i 112. (Contributed by Wolf Lammen, 12-May-2013.)
𝜑    &   𝜓 → ¬ 𝜑)       𝜓

Theorempm2.21i 115 A contradiction implies anything. Inference associated with pm2.21 119. Its associated inference is pm2.24ii 116. (Contributed by NM, 16-Sep-1993.)
¬ 𝜑       (𝜑𝜓)

Theorempm2.24ii 116 A contradiction implies anything. Inference associated with pm2.21i 115 and pm2.24i 145. (Contributed by NM, 27-Feb-2008.)
𝜑    &    ¬ 𝜑       𝜓

Theorempm2.21d 117 A contradiction implies anything. Deduction associated with pm2.21 119. (Contributed by NM, 10-Feb-1996.)
(𝜑 → ¬ 𝜓)       (𝜑 → (𝜓𝜒))

Theorempm2.21ddALT 118 Alternate proof of pm2.21dd 185. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   (𝜑 → ¬ 𝜓)       (𝜑𝜒)

Theorempm2.21 119 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 115. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 14-Sep-2012.)
𝜑 → (𝜑𝜓))

Theorempm2.24 120 Theorem *2.24 of [WhiteheadRussell] p. 104. Its associated inference is pm2.24i 145. (Contributed by NM, 3-Jan-2005.)
(𝜑 → (¬ 𝜑𝜓))

Theorempm2.18 121 Proof by contradiction. Theorem *2.18 of [WhiteheadRussell] p. 103. Also called the Law of Clavius. See also pm2.01 179. (Contributed by NM, 29-Dec-1992.)
((¬ 𝜑𝜑) → 𝜑)

Theorempm2.18i 122 Inference associated with pm2.18 121. (Contributed by BJ, 30-Mar-2020.)
𝜑𝜑)       𝜑

Theorempm2.18d 123 Deduction based on reductio ad absurdum. (Contributed by FL, 12-Jul-2009.) (Proof shortened by Andrew Salmon, 7-May-2011.)
(𝜑 → (¬ 𝜓𝜓))       (𝜑𝜓)

Theoremnotnotr 124 Double negation elimination. Converse of notnot 135 and one implication of notnotb 303. 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 125 Inference associated with notnotr 124. (Contributed by NM, 27-Feb-2008.) (Proof shortened by Wolf Lammen, 15-Jul-2021.)
¬ ¬ 𝜑       𝜑

TheoremnotnotriOLD 126 Obsolete proof of notnotri 125 as of 15-Jul-2021 . (Contributed by NM, 27-Feb-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ ¬ 𝜑       𝜑

Theoremnotnotrd 127 Deduction associated with notnotr 124 and notnotri 125. Double negation elimination rule. A translation of the natural deduction rule ¬ ¬ C , Γ¬ ¬ 𝜓 ⇒ Γ𝜓; see natded 26652. 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 128 A contraposition deduction. (Contributed by NM, 19-Aug-1993.)
(𝜑 → (𝜓 → ¬ 𝜒))       (𝜑 → (𝜒 → ¬ 𝜓))

Theoremcon2 129 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 130 Modus tollens deduction. (Contributed by NM, 4-Jul-1994.)
(𝜑𝜒)    &   (𝜑 → (𝜓 → ¬ 𝜒))       (𝜑 → ¬ 𝜓)

Theoremmt2i 131 Modus tollens inference. (Contributed by NM, 26-Mar-1995.) (Proof shortened by Wolf Lammen, 15-Sep-2012.)
𝜒    &   (𝜑 → (𝜓 → ¬ 𝜒))       (𝜑 → ¬ 𝜓)

Theoremnsyl3 132 A negated syllogism inference. (Contributed by NM, 1-Dec-1995.)
(𝜑 → ¬ 𝜓)    &   (𝜒𝜓)       (𝜒 → ¬ 𝜑)

Theoremcon2i 133 A contraposition inference. Its associated inference is mt2 190. (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 134 A negated syllogism inference. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Wolf Lammen, 2-Mar-2013.)
(𝜑 → ¬ 𝜓)    &   (𝜒𝜓)       (𝜑 → ¬ 𝜒)

Theoremnotnot 135 Double negation introduction. Converse of notnotr 124 and one implication of notnotb 303. 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 136 Inference associated with notnot 135. (Contributed by NM, 27-Feb-2008.)
𝜑        ¬ ¬ 𝜑

Theoremnotnotd 137 Deduction associated with notnot 135 and notnoti 136. (Contributed by Jarvin Udandy, 2-Sep-2016.) Avoid biconditional. (Revised by Wolf Lammen, 27-Mar-2021.)
(𝜑𝜓)       (𝜑 → ¬ ¬ 𝜓)

Theoremcon1d 138 A contraposition deduction. (Contributed by NM, 27-Dec-1992.)
(𝜑 → (¬ 𝜓𝜒))       (𝜑 → (¬ 𝜒𝜓))

Theoremmt3d 139 Modus tollens deduction. (Contributed by NM, 26-Mar-1995.)
(𝜑 → ¬ 𝜒)    &   (𝜑 → (¬ 𝜓𝜒))       (𝜑𝜓)

Theoremmt3i 140 Modus tollens inference. (Contributed by NM, 26-Mar-1995.) (Proof shortened by Wolf Lammen, 15-Sep-2012.)
¬ 𝜒    &   (𝜑 → (¬ 𝜓𝜒))       (𝜑𝜓)

Theoremnsyl2 141 A negated syllogism inference. (Contributed by NM, 26-Jun-1994.)
(𝜑 → ¬ 𝜓)    &   𝜒𝜓)       (𝜑𝜒)

Theoremcon1 142 Contraposition. Theorem *2.15 of [WhiteheadRussell] p. 102. Its associated inference is con1i 143. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 12-Feb-2013.)
((¬ 𝜑𝜓) → (¬ 𝜓𝜑))

Theoremcon1i 143 A contraposition inference. Inference associated with con1 142. Its associated inference is mt3 191. (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 144 Obsolete proof of con4i 112 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 145 Inference associated with pm2.24 120. Its associated inference is pm2.24ii 116. (Contributed by NM, 20-Aug-2001.)
𝜑       𝜑𝜓)

Theorempm2.24d 146 Deduction form of pm2.24 120. (Contributed by NM, 30-Jan-2006.)
(𝜑𝜓)       (𝜑 → (¬ 𝜓𝜒))

Theoremcon3d 147 A contraposition deduction. Deduction form of con3 148. (Contributed by NM, 10-Jan-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → (¬ 𝜒 → ¬ 𝜓))

Theoremcon3 148 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 149. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 13-Feb-2013.)
((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))

Theoremcon3i 149 A contraposition inference. Inference associated with con3 148. Its associated inference is mto 187. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 20-Jun-2013.)
(𝜑𝜓)       𝜓 → ¬ 𝜑)

Theoremcon3rr3 150 Rotate through consequent right. (Contributed by Wolf Lammen, 3-Nov-2013.)
(𝜑 → (𝜓𝜒))       𝜒 → (𝜑 → ¬ 𝜓))

Theoremmt4d 151 Modus tollens deduction. Deduction form of mt4 114. (Contributed by NM, 9-Jun-2006.)
(𝜑𝜓)    &   (𝜑 → (¬ 𝜒 → ¬ 𝜓))       (𝜑𝜒)

Theoremmt4i 152 Modus tollens inference. (Contributed by Wolf Lammen, 12-May-2013.)
𝜒    &   (𝜑 → (¬ 𝜓 → ¬ 𝜒))       (𝜑𝜓)

Theoremnsyld 153 A negated syllogism deduction. (Contributed by NM, 9-Apr-2005.)
(𝜑 → (𝜓 → ¬ 𝜒))    &   (𝜑 → (𝜏𝜒))       (𝜑 → (𝜓 → ¬ 𝜏))

Theoremnsyli 154 A negated syllogism inference. (Contributed by NM, 3-May-1994.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → ¬ 𝜒)       (𝜑 → (𝜃 → ¬ 𝜓))

Theoremnsyl4 155 A negated syllogism inference. (Contributed by NM, 15-Feb-1996.)
(𝜑𝜓)    &   𝜑𝜒)       𝜒𝜓)

Theorempm3.2im 156 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 157 Theorem 8 of [Margaris] p. 60. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.)
(𝜑 → (¬ 𝜓 → ¬ (𝜑𝜓)))

Theoremjc 158 Deduction joining the consequents of two premises. A deduction associated with pm3.2im 156. (Contributed by NM, 28-Dec-1992.)
(𝜑𝜓)    &   (𝜑𝜒)       (𝜑 → ¬ (𝜓 → ¬ 𝜒))

Theoremimpi 159 An importation inference. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 20-Jul-2013.)
(𝜑 → (𝜓𝜒))       (¬ (𝜑 → ¬ 𝜓) → 𝜒)

Theoremexpi 160 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 161 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 162 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)

Theorempm2.5 163 Theorem *2.5 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 9-Oct-2012.)
(¬ (𝜑𝜓) → (¬ 𝜑𝜓))

Theorempm2.51 164 Theorem *2.51 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
(¬ (𝜑𝜓) → (𝜑 → ¬ 𝜓))

Theorempm2.521 165 Theorem *2.521 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 8-Oct-2012.)
(¬ (𝜑𝜓) → (𝜓𝜑))

Theorempm2.52 166 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 167 Exportation theorem expressed with primitive connectives. (Contributed by NM, 28-Dec-1992.)
((¬ (𝜑 → ¬ 𝜓) → 𝜒) → (𝜑 → (𝜓𝜒)))

Theoremimpt 168 Importation theorem 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 169 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 170 Inference eliminating an antecedent. (Contributed by NM, 15-Jul-2005.)
(𝜑 → (𝜓𝜒))    &   𝜓𝜒)       (𝜑𝜒)

Theorempm2.61d2 171 Inference eliminating an antecedent. (Contributed by NM, 18-Aug-1993.)
(𝜑 → (¬ 𝜓𝜒))    &   (𝜓𝜒)       (𝜑𝜒)

Theoremja 172 Inference joining the antecedents of two premises. For partial converses, see jarr 104 and jarl 174. (Contributed by NM, 24-Jan-1993.) (Proof shortened by Mel L. O'Cat, 19-Feb-2008.)
𝜑𝜒)    &   (𝜓𝜒)       ((𝜑𝜓) → 𝜒)

Theoremjad 173 Deduction form of ja 172. (Contributed by Scott Fenton, 13-Dec-2010.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
(𝜑 → (¬ 𝜓𝜃))    &   (𝜑 → (𝜒𝜃))       (𝜑 → ((𝜓𝜒) → 𝜃))

Theoremjarl 174 Elimination of a nested antecedent as a partial converse of ja 172 (the other being jarr 104). (Contributed by Wolf Lammen, 10-May-2013.)
(((𝜑𝜓) → 𝜒) → (¬ 𝜑𝜒))

Theorempm2.61i 175 Inference eliminating an antecedent. (Contributed by NM, 5-Apr-1994.) (Proof shortened by Wolf Lammen, 12-Sep-2013.)
(𝜑𝜓)    &   𝜑𝜓)       𝜓

Theorempm2.61ii 176 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 177 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 178 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

Theorempm2.01 179 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 pm2.18 121. (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 180 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)

Theorempm2.6 181 Theorem *2.6 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
((¬ 𝜑𝜓) → ((𝜑𝜓) → 𝜓))

Theorempm2.61 182 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))

Theorempm2.65 183 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 184 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 185 A contradiction implies anything. Deduction from pm2.21 119. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof shortened by Wolf Lammen, 22-Jul-2019.)
(𝜑𝜓)    &   (𝜑 → ¬ 𝜓)       (𝜑𝜒)

Theorempm2.65d 186 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 187 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 149. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)
¬ 𝜓    &   (𝜑𝜓)        ¬ 𝜑

Theoremmtod 188 Modus tollens deduction. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)
(𝜑 → ¬ 𝜒)    &   (𝜑 → (𝜓𝜒))       (𝜑 → ¬ 𝜓)

Theoremmtoi 189 Modus tollens inference. (Contributed by NM, 5-Jul-1994.) (Proof shortened by Wolf Lammen, 15-Sep-2012.)
¬ 𝜒    &   (𝜑 → (𝜓𝜒))       (𝜑 → ¬ 𝜓)

Theoremmt2 190 A rule similar to modus tollens. Inference associated with con2i 133. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 10-Sep-2013.)
𝜓    &   (𝜑 → ¬ 𝜓)        ¬ 𝜑

Theoremmt3 191 A rule similar to modus tollens. Inference associated with con1i 143. (Contributed by NM, 18-May-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)
¬ 𝜓    &   𝜑𝜓)       𝜑

Theorempeirce 192 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 193 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 83. (Contributed by NM, 12-Aug-2004.)
(((𝜑𝜓) → 𝜓) → ((𝜓𝜑) → 𝜑))

Theorembijust 194 Theorem used to justify definition of biconditional df-bi 196. (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 196 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 195 Extend our wff definition to include the biconditional connective.
wff (𝜑𝜓)

Definitiondf-bi 196 Define the biconditional (logical 'iff').

The definition df-bi 196 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 1033) 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 194. It is impossible to use df-bi 196 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 196 in the proof with the corresponding bijust 194 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 1478) and the constant false (df-fal 1481), we will be able to prove these truth table values: ((⊤ ↔ ⊤) ↔ ⊤) (trubitru 1511), ((⊤ ↔ ⊥) ↔ ⊥) (trubifal 1513), ((⊥ ↔ ⊤) ↔ ⊥) (falbitru 1512), and ((⊥ ↔ ⊥) ↔ ⊤) (falbifal 1514).

See dfbi1 202, dfbi2 658, and dfbi3 933 for theorems suggesting typical textbook definitions of , showing that our definition has the properties we expect. Theorem dfbi1 202 is particularly useful if we want to eliminate from an expression to convert it to primitives. Theorem dfbi 659 shows this definition rewritten in an abbreviated form after conjunction is introduced, for easier understanding.

Contrast with (df-or 384), (wi 4), (df-nan 1440), and (df-xor 1457) . In some sense returns true if two truth values are equal; = (df-cleq 2603) returns true if two classes are equal. (Contributed by NM, 27-Dec-1992.)

¬ (((𝜑𝜓) → ¬ ((𝜑𝜓) → ¬ (𝜓𝜑))) → ¬ (¬ ((𝜑𝜓) → ¬ (𝜓𝜑)) → (𝜑𝜓)))

Theoremimpbi 197 Property of the biconditional connective. (Contributed by NM, 11-May-1999.)
((𝜑𝜓) → ((𝜓𝜑) → (𝜑𝜓)))

Theoremimpbii 198 Infer an equivalence from an implication and its converse. Inference associated with impbi 197. (Contributed by NM, 29-Dec-1992.)
(𝜑𝜓)    &   (𝜓𝜑)       (𝜑𝜓)

Theoremimpbidd 199 Deduce an equivalence from two implications. Double deduction associated with impbi 197 and impbii 198. Deduction associated with impbid 201. (Contributed by Rodolfo Medina, 12-Oct-2010.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜑 → (𝜓 → (𝜃𝜒)))       (𝜑 → (𝜓 → (𝜒𝜃)))

Theoremimpbid21d 200 Deduce an equivalence from two implications. (Contributed by Wolf Lammen, 12-May-2013.)
(𝜓 → (𝜒𝜃))    &   (𝜑 → (𝜃𝜒))       (𝜑 → (𝜓 → (𝜒𝜃)))

Page List
Jump to page: Contents  1 1-100101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42360
 Copyright terms: Public domain < Previous  Next >