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Theorem List for Metamath Proof Explorer - 101-200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempm2.21dd 101 A contradiction implies anything. Deduction from pm2.21 102. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  -.  ps )   =>    |-  ( ph  ->  ch )
 
Theorempm2.21 102 From a wff and its negation, anything is true. Theorem *2.21 of [WhiteheadRussell] p. 104. Also called the Duns Scotus law. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Sep-2012.)
 |-  ( -.  ph  ->  (
 ph  ->  ps ) )
 
Theorempm2.24 103 Theorem *2.24 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
 |-  ( ph  ->  ( -.  ph  ->  ps )
 )
 
Theorempm2.18 104 Proof by contradiction. Theorem *2.18 of [WhiteheadRussell] p. 103. Also called the Law of Clavius. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( -.  ph  -> 
 ph )  ->  ph )
 
Theorempm2.18d 105 Deduction based on reductio ad absurdum. (Contributed by FL, 12-Jul-2009.) (Proof shortened by Andrew Salmon, 7-May-2011.)
 |-  ( ph  ->  ( -.  ps  ->  ps )
 )   =>    |-  ( ph  ->  ps )
 
Theoremnotnot2 106 Converse of double negation. Theorem *2.14 of [WhiteheadRussell] p. 102. (Contributed by NM, 5-Aug-1993.) (Proof shortened by David Harvey, 5-Sep-1999.) (Proof shortened by Josh Purinton, 29-Dec-2000.)
 |-  ( -.  -.  ph  -> 
 ph )
 
Theoremnotnotrd 107 Deduction converting double-negation into the original wff, aka the double negation rule. A translation of natural deduction rule  -.  -. -C,  _G |-  -.  -.  ps =>  _G |-  ps; see natded 4. This is definition NNC in [Pfenning] p. 17. This rule is valid in classical logic (which MPE uses), but not intuitionistic logic. (Contributed by DAW, 8-Feb-2017.)
 |-  ( ph  ->  -.  -.  ps )   =>    |-  ( ph  ->  ps )
 
Theoremnotnotri 108 Inference from double negation. (Contributed by NM, 27-Feb-2008.)
 |- 
 -.  -.  ph   =>    |-  ph
 
Theoremcon2d 109 A contraposition deduction. (Contributed by NM, 19-Aug-1993.)
 |-  ( ph  ->  ( ps  ->  -.  ch )
 )   =>    |-  ( ph  ->  ( ch  ->  -.  ps )
 )
 
Theoremcon2 110 Contraposition. Theorem *2.03 of [WhiteheadRussell] p. 100. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Feb-2013.)
 |-  ( ( ph  ->  -. 
 ps )  ->  ( ps  ->  -.  ph ) )
 
Theoremmt2d 111 Modus tollens deduction. (Contributed by NM, 4-Jul-1994.)
 |-  ( ph  ->  ch )   &    |-  ( ph  ->  ( ps  ->  -. 
 ch ) )   =>    |-  ( ph  ->  -. 
 ps )
 
Theoremmt2i 112 Modus tollens inference. (Contributed by NM, 26-Mar-1995.) (Proof shortened by Wolf Lammen, 15-Sep-2012.)
 |- 
 ch   &    |-  ( ph  ->  ( ps  ->  -.  ch )
 )   =>    |-  ( ph  ->  -.  ps )
 
Theoremnsyl3 113 A negated syllogism inference. (Contributed by NM, 1-Dec-1995.)
 |-  ( ph  ->  -.  ps )   &    |-  ( ch  ->  ps )   =>    |-  ( ch  ->  -.  ph )
 
Theoremcon2i 114 A contraposition inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 28-Nov-2008.) (Proof shortened by Wolf Lammen, 13-Jun-2013.)
 |-  ( ph  ->  -.  ps )   =>    |-  ( ps  ->  -.  ph )
 
Theoremnsyl 115 A negated syllogism inference. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Wolf Lammen, 2-Mar-2013.)
 |-  ( ph  ->  -.  ps )   &    |-  ( ch  ->  ps )   =>    |-  ( ph  ->  -.  ch )
 
Theoremnotnot1 116 Converse of double negation. Theorem *2.12 of [WhiteheadRussell] p. 101. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 2-Mar-2013.)
 |-  ( ph  ->  -.  -.  ph )
 
Theoremnotnoti 117 Infer double negation. (Contributed by NM, 27-Feb-2008.)
 |-  ph   =>    |- 
 -.  -.  ph
 
Theoremcon1d 118 A contraposition deduction. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( -.  ps  ->  ch )
 )   =>    |-  ( ph  ->  ( -.  ch  ->  ps )
 )
 
Theoremmt3d 119 Modus tollens deduction. (Contributed by NM, 26-Mar-1995.)
 |-  ( ph  ->  -.  ch )   &    |-  ( ph  ->  ( -.  ps  ->  ch )
 )   =>    |-  ( ph  ->  ps )
 
Theoremmt3i 120 Modus tollens inference. (Contributed by NM, 26-Mar-1995.) (Proof shortened by Wolf Lammen, 15-Sep-2012.)
 |- 
 -.  ch   &    |-  ( ph  ->  ( -.  ps  ->  ch )
 )   =>    |-  ( ph  ->  ps )
 
Theoremnsyl2 121 A negated syllogism inference. (Contributed by NM, 26-Jun-1994.)
 |-  ( ph  ->  -.  ps )   &    |-  ( -.  ch  ->  ps )   =>    |-  ( ph  ->  ch )
 
Theoremcon1 122 Contraposition. Theorem *2.15 of [WhiteheadRussell] p. 102. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Feb-2013.)
 |-  ( ( -.  ph  ->  ps )  ->  ( -.  ps  ->  ph ) )
 
Theoremcon1i 123 A contraposition inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 28-Nov-2008.) (Proof shortened by Wolf Lammen, 19-Jun-2013.)
 |-  ( -.  ph  ->  ps )   =>    |-  ( -.  ps  ->  ph )
 
Theoremcon4i 124 Inference rule derived from axiom ax-3 9. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 21-Jun-2013.)
 |-  ( -.  ph  ->  -. 
 ps )   =>    |-  ( ps  ->  ph )
 
Theorempm2.21i 125 A contradiction implies anything. Inference from pm2.21 102. (Contributed by NM, 16-Sep-1993.)
 |- 
 -.  ph   =>    |-  ( ph  ->  ps )
 
Theorempm2.24ii 126 A contradiction implies anything. Inference from pm2.24 103. (Contributed by NM, 27-Feb-2008.)
 |-  ph   &    |- 
 -.  ph   =>    |- 
 ps
 
Theoremcon3d 127 A contraposition deduction. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( -.  ch  ->  -.  ps ) )
 
Theoremcon3 128 Contraposition. Theorem *2.16 of [WhiteheadRussell] p. 103. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-Feb-2013.)
 |-  ( ( ph  ->  ps )  ->  ( -.  ps 
 ->  -.  ph ) )
 
Theoremcon3i 129 A contraposition inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 20-Jun-2013.)
 |-  ( ph  ->  ps )   =>    |-  ( -.  ps  ->  -.  ph )
 
Theoremcon3rr3 130 Rotate through consequent right. (Contributed by Wolf Lammen, 3-Nov-2013.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( -.  ch  ->  ( ph  ->  -.  ps ) )
 
Theoremmt4 131 The rule of modus tollens. (Contributed by Wolf Lammen, 12-May-2013.)
 |-  ph   &    |-  ( -.  ps  ->  -.  ph )   =>    |- 
 ps
 
Theoremmt4d 132 Modus tollens deduction. (Contributed by NM, 9-Jun-2006.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ( -.  ch  ->  -.  ps ) )   =>    |-  ( ph  ->  ch )
 
Theoremmt4i 133 Modus tollens inference. (Contributed by Wolf Lammen, 12-May-2013.)
 |- 
 ch   &    |-  ( ph  ->  ( -.  ps  ->  -.  ch )
 )   =>    |-  ( ph  ->  ps )
 
Theoremnsyld 134 A negated syllogism deduction. (Contributed by NM, 9-Apr-2005.)
 |-  ( ph  ->  ( ps  ->  -.  ch )
 )   &    |-  ( ph  ->  ( ta  ->  ch ) )   =>    |-  ( ph  ->  ( ps  ->  -.  ta )
 )
 
Theoremnsyli 135 A negated syllogism inference. (Contributed by NM, 3-May-1994.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( th  ->  -.  ch )   =>    |-  ( ph  ->  ( th  ->  -. 
 ps ) )
 
Theoremnsyl4 136 A negated syllogism inference. (Contributed by NM, 15-Feb-1996.)
 |-  ( ph  ->  ps )   &    |-  ( -.  ph  ->  ch )   =>    |-  ( -.  ch  ->  ps )
 
Theorempm2.24d 137 Deduction version of pm2.24 103. (Contributed by NM, 30-Jan-2006.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( -.  ps  ->  ch ) )
 
Theorempm2.24i 138 Inference version of pm2.24 103. (Contributed by NM, 20-Aug-2001.)
 |-  ph   =>    |-  ( -.  ph  ->  ps )
 
Theorempm3.2im 139 Theorem *3.2 of [WhiteheadRussell] p. 111, expressed with primitive connectives. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.)
 |-  ( ph  ->  ( ps  ->  -.  ( ph  ->  -.  ps ) ) )
 
Theoremmth8 140 Theorem 8 of [Margaris] p. 60. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.)
 |-  ( ph  ->  ( -.  ps  ->  -.  ( ph  ->  ps ) ) )
 
Theoremjc 141 Inference joining the consequents of two premises. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   =>    |-  ( ph  ->  -.  ( ps  ->  -.  ch )
 )
 
Theoremimpi 142 An importation inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 20-Jul-2013.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( -.  ( ph  ->  -.  ps )  ->  ch )
 
Theoremexpi 143 An exportation inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 28-Nov-2008.)
 |-  ( -.  ( ph  ->  -.  ps )  ->  ch )   =>    |-  ( ph  ->  ( ps  ->  ch ) )
 
Theoremsimprim 144 Simplification. Similar to Theorem *3.27 (Simp) of [WhiteheadRussell] p. 112. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-Nov-2012.)
 |-  ( -.  ( ph  ->  -.  ps )  ->  ps )
 
Theoremsimplim 145 Simplification. Similar to Theorem *3.26 (Simp) of [WhiteheadRussell] p. 112. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 21-Jul-2012.)
 |-  ( -.  ( ph  ->  ps )  ->  ph )
 
Theorempm2.5 146 Theorem *2.5 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 9-Oct-2012.)
 |-  ( -.  ( ph  ->  ps )  ->  ( -.  ph  ->  ps )
 )
 
Theorempm2.51 147 Theorem *2.51 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
 |-  ( -.  ( ph  ->  ps )  ->  ( ph  ->  -.  ps )
 )
 
Theorempm2.521 148 Theorem *2.521 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 8-Oct-2012.)
 |-  ( -.  ( ph  ->  ps )  ->  ( ps  ->  ph ) )
 
Theorempm2.52 149 Theorem *2.52 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 8-Oct-2012.)
 |-  ( -.  ( ph  ->  ps )  ->  ( -.  ph  ->  -.  ps )
 )
 
Theoremexpt 150 Exportation theorem expressed with primitive connectives. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( -.  ( ph  ->  -.  ps )  ->  ch )  ->  ( ph  ->  ( ps  ->  ch ) ) )
 
Theoremimpt 151 Importation theorem expressed with primitive connectives. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 20-Jul-2013.)
 |-  ( ( ph  ->  ( ps  ->  ch )
 )  ->  ( -.  ( ph  ->  -.  ps )  ->  ch ) )
 
Theorempm2.61d 152 Deduction eliminating an antecedent. (Contributed by NM, 27-Apr-1994.) (Proof shortened by Wolf Lammen, 12-Sep-2013.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( -.  ps  ->  ch ) )   =>    |-  ( ph  ->  ch )
 
Theorempm2.61d1 153 Inference eliminating an antecedent. (Contributed by NM, 15-Jul-2005.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( -.  ps  ->  ch )   =>    |-  ( ph  ->  ch )
 
Theorempm2.61d2 154 Inference eliminating an antecedent. (Contributed by NM, 18-Aug-1993.)
 |-  ( ph  ->  ( -.  ps  ->  ch )
 )   &    |-  ( ps  ->  ch )   =>    |-  ( ph  ->  ch )
 
Theoremja 155 Inference joining the antecedents of two premises. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 19-Feb-2008.)
 |-  ( -.  ph  ->  ch )   &    |-  ( ps  ->  ch )   =>    |-  ( ( ph  ->  ps )  ->  ch )
 
Theoremjad 156 Deduction form of ja 155. (Contributed by Scott Fenton, 13-Dec-2010.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 |-  ( ph  ->  ( -.  ps  ->  th )
 )   &    |-  ( ph  ->  ( ch  ->  th ) )   =>    |-  ( ph  ->  ( ( ps  ->  ch )  ->  th ) )
 
Theoremjarl 157 Elimination of a nested antecedent as a kind of reversal of inference ja 155. (Contributed by Wolf Lammen, 10-May-2013.)
 |-  ( ( ( ph  ->  ps )  ->  ch )  ->  ( -.  ph  ->  ch ) )
 
Theorempm2.61i 158 Inference eliminating an antecedent. (Contributed by NM, 5-Apr-1994.) (Proof shortened by Wolf Lammen, 12-Sep-2013.)
 |-  ( ph  ->  ps )   &    |-  ( -.  ph  ->  ps )   =>    |-  ps
 
Theorempm2.61ii 159 Inference eliminating two antecedents. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.)
 |-  ( -.  ph  ->  ( -.  ps  ->  ch )
 )   &    |-  ( ph  ->  ch )   &    |-  ( ps  ->  ch )   =>    |- 
 ch
 
Theorempm2.61nii 160 Inference eliminating two antecedents. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 13-Nov-2012.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( -.  ph  ->  ch )   &    |-  ( -.  ps  ->  ch )   =>    |-  ch
 
Theorempm2.61iii 161 Inference eliminating three antecedents. (Contributed by NM, 2-Jan-2002.) (Proof shortened by Wolf Lammen, 22-Sep-2013.)
 |-  ( -.  ph  ->  ( -.  ps  ->  ( -.  ch  ->  th )
 ) )   &    |-  ( ph  ->  th )   &    |-  ( ps  ->  th )   &    |-  ( ch  ->  th )   =>    |- 
 th
 
Theorempm2.01 162 Reductio ad absurdum. Theorem *2.01 of [WhiteheadRussell] p. 100. (Contributed by NM, 18-Aug-1993.) (Proof shortened by O'Cat, 21-Nov-2008.) (Proof shortened by Wolf Lammen, 31-Oct-2012.)
 |-  ( ( ph  ->  -.  ph )  ->  -.  ph )
 
Theorempm2.01d 163 Deduction based on reductio ad absurdum. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 5-Mar-2013.)
 |-  ( ph  ->  ( ps  ->  -.  ps )
 )   =>    |-  ( ph  ->  -.  ps )
 
Theorempm2.6 164 Theorem *2.6 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( -.  ph  ->  ps )  ->  (
 ( ph  ->  ps )  ->  ps ) )
 
Theorempm2.61 165 Theorem *2.61 of [WhiteheadRussell] p. 107. Useful for eliminating an antecedent. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 22-Sep-2013.)
 |-  ( ( ph  ->  ps )  ->  ( ( -.  ph  ->  ps )  ->  ps ) )
 
Theorempm2.65 166 Theorem *2.65 of [WhiteheadRussell] p. 107. Proof by contradiction. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 8-Mar-2013.)
 |-  ( ( ph  ->  ps )  ->  ( ( ph  ->  -.  ps )  ->  -.  ph ) )
 
Theorempm2.65i 167 Inference rule for proof by contradiction. (Contributed by NM, 18-May-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  -.  ps )   =>    |-  -.  ph
 
Theorempm2.65d 168 Deduction rule for proof by contradiction. (Contributed by NM, 26-Jun-1994.) (Proof shortened by Wolf Lammen, 26-May-2013.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ps  ->  -. 
 ch ) )   =>    |-  ( ph  ->  -. 
 ps )
 
Theoremmto 169 The rule of modus tollens. The rule says, "if  ps is not true, and  ph implies  ps, then  ps must also be not true." Modus tollens is short for "modus tollendo tollens," a Latin phrase that means "the mood that by denying affirms" [Sanford] p. 39. It is also called denying the consequent. Modus tollens is closely related to modus ponens ax-mp 10. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)
 |- 
 -.  ps   &    |-  ( ph  ->  ps )   =>    |- 
 -.  ph
 
Theoremmtod 170 Modus tollens deduction. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)
 |-  ( ph  ->  -.  ch )   &    |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  -. 
 ps )
 
Theoremmtoi 171 Modus tollens inference. (Contributed by NM, 5-Jul-1994.) (Proof shortened by Wolf Lammen, 15-Sep-2012.)
 |- 
 -.  ch   &    |-  ( ph  ->  ( ps  ->  ch )
 )   =>    |-  ( ph  ->  -.  ps )
 
Theoremmt2 172 A rule similar to modus tollens. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 10-Sep-2013.)
 |- 
 ps   &    |-  ( ph  ->  -.  ps )   =>    |- 
 -.  ph
 
Theoremmt3 173 A rule similar to modus tollens. (Contributed by NM, 18-May-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)
 |- 
 -.  ps   &    |-  ( -.  ph  ->  ps )   =>    |-  ph
 
Theorempeirce 174 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 7 through ax-3 9. A curious fact about this theorem is that it requires ax-3 9 for its proof even though the result has no negation connectives in it. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 9-Oct-2012.)
 |-  ( ( ( ph  ->  ps )  ->  ph )  -> 
 ph )
 
Theoremloolin 175 The Linearity Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz. For a version not using ax-3 9, see loolinALT 97. (Contributed by O'Cat, 12-Aug-2004.) (Proof shortened by Wolf Lammen, 2-Nov-2012.)
 |-  ( ( ( ph  ->  ps )  ->  ( ps  ->  ph ) )  ->  ( ps  ->  ph ) )
 
Theoremlooinv 176 The Inversion Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz. Using dfor2 402, we can see that this essentially expresses "disjunction commutes." Theorem *2.69 of [WhiteheadRussell] p. 108. (Contributed by NM, 12-Aug-2004.)
 |-  ( ( ( ph  ->  ps )  ->  ps )  ->  ( ( ps  ->  ph )  ->  ph ) )
 
Theorembijust 177 Theorem used to justify definition of biconditional df-bi 179. (Contributed by NM, 11-May-1999.) (Proof shortened by Josh Purinton, 29-Dec-2000.)
 |- 
 -.  ( ( -.  ( ( ph  ->  ps )  ->  -.  ( ps  ->  ph ) )  ->  -.  ( ( ph  ->  ps )  ->  -.  ( ps  ->  ph ) ) ) 
 ->  -.  ( -.  (
 ( ph  ->  ps )  ->  -.  ( ps  ->  ph ) )  ->  -.  (
 ( ph  ->  ps )  ->  -.  ( ps  ->  ph ) ) ) )
 
1.3.5  Logical equivalence

The definition df-bi 179 in this section is our first definition, which introduces and defines the biconditional connective  <->. We define a wff of the form  ( ph  <->  ps ) as an abbreviation for  -.  ( (
ph  ->  ps )  ->  -.  ( ps  ->  ph )
).

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 361 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 178 Extend our wff definition to include the biconditional connective.
 wff  ( ph  <->  ps )
 
Definitiondf-bi 179 Define the biconditional (logical 'iff').

The definition df-bi 179 in this section is our first definition, which introduces and defines the biconditional connective  <->. We define a wff of the form  ( ph  <->  ps ) as an abbreviation for  -.  ( (
ph  ->  ps )  ->  -.  ( ps  ->  ph )
).

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 361 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 941) 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  ( ph  <->  ps ) (the definiendum i.e. the thing being defined) with  -.  ( (
ph  ->  ps )  ->  -.  ( ps  ->  ph )
) (the definiens i.e. the defining expression) in the definition, the definition becomes the previously proved theorem bijust 177. It is impossible to use df-bi 179 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 179 in the proof with the corresponding bijust 177 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 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  T. (df-tru 1315) and the constant false  F. (df-fal 1316), we will be able to prove these truth table values:  ( (  T.  <->  T.  )  <->  T.  ) (trubitru 1346),  ( (  T. 
<->  F.  )  <->  F.  ) (trubifal 1347), 
( (  F.  <->  T.  )  <->  F.  ) (falbitru 1348), and  ( (  F.  <->  F.  )  <->  T.  ) (falbifal 1349).

See dfbi1 186, dfbi2 612, and dfbi3 868 for theorems suggesting typical textbook definitions of  <->, showing that our definition has the properties we expect. Theorem dfbi1 186 is particularly useful if we want to eliminate 
<-> from an expression to convert it to primitives. Theorem dfbi 613 shows this definition rewritten in an abbreviated form after conjunction is introduced, for easier understanding.

Contrast with  \/ (df-or 361), 
-> (wi 6),  -/\ (df-nan 1293), and  \/_ (df-xor 1301) . In some sense  <-> returns true if two truth values are equal;  = (df-cleq 2251) returns true if two classes are equal. (Contributed by NM, 5-Aug-1993.)

 |- 
 -.  ( ( (
 ph 
 <->  ps )  ->  -.  (
 ( ph  ->  ps )  ->  -.  ( ps  ->  ph ) ) )  ->  -.  ( -.  ( (
 ph  ->  ps )  ->  -.  ( ps  ->  ph ) )  ->  ( ph  <->  ps ) ) )
 
Theorembi1 180 Property of the biconditional connective. (Contributed by NM, 11-May-1999.)
 |-  ( ( ph  <->  ps )  ->  ( ph  ->  ps ) )
 
Theorembi3 181 Property of the biconditional connective. (Contributed by NM, 11-May-1999.)
 |-  ( ( ph  ->  ps )  ->  ( ( ps  ->  ph )  ->  ( ph 
 <->  ps ) ) )
 
Theoremimpbii 182 Infer an equivalence from an implication and its converse. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ps )   &    |-  ( ps  ->  ph )   =>    |-  ( ph  <->  ps )
 
Theoremimpbidd 183 Deduce an equivalence from two implications. (Contributed by Rodolfo Medina, 12-Oct-2010.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )   &    |-  ( ph  ->  ( ps  ->  ( th  ->  ch )
 ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  <->  th ) ) )
 
Theoremimpbid21d 184 Deduce an equivalence from two implications. (Contributed by Wolf Lammen, 12-May-2013.)
 |-  ( ps  ->  ( ch  ->  th ) )   &    |-  ( ph  ->  ( th  ->  ch ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  <->  th ) ) )
 
Theoremimpbid 185 Deduce an equivalence from two implications. (Contributed by NM, 5-Aug-1993.) (Revised by Wolf Lammen, 3-Nov-2012.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ch  ->  ps ) )   =>    |-  ( ph  ->  ( ps 
 <->  ch ) )
 
Theoremdfbi1 186 Relate the biconditional connective to primitive connectives. See dfbi1gb 187 for an unusual version proved directly from axioms. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( ph  <->  ps )  <->  -.  ( ( ph  ->  ps )  ->  -.  ( ps  ->  ph ) ) )
 
Theoremdfbi1gb 187 This proof of dfbi1 186, discovered by Gregory Bush on 8-Mar-2004, has several curious properties. First, it has only 17 steps directly from the axioms and df-bi 179, compared to over 800 steps were the proof of dfbi1 186 expanded into axioms. Second, step 2 demands only the property of "true"; any axiom (or theorem) could be used. It might be thought, therefore, that it is in some sense redundant, but in fact no proof is shorter than this (measured by number of steps). Third, it illustrates how intermediate steps can "blow up" in size even in short proofs. Fourth, the compressed proof is only 182 bytes (or 17 bytes in D-proof notation), but the generated web page is over 200kB with intermediate steps that are essentially incomprehensible to humans (other than Gregory Bush). If there were an obfuscated code contest for proofs, this would be a contender. This "blowing up" and incomprehensibility of the intermediate steps vividly demonstrate the advantages of using many layered intermediate theorems, since each theorem is easier to understand. (Contributed by Gregory Bush, 10-Mar-2004.) (Proof modification is discouraged.)
 |-  ( ( ph  <->  ps )  <->  -.  ( ( ph  ->  ps )  ->  -.  ( ps  ->  ph ) ) )
 
Theorembiimpi 188 Infer an implication from a logical equivalence. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  ps )   =>    |-  ( ph  ->  ps )
 
Theoremsylbi 189 A mixed syllogism inference from a biconditional and an implication. Useful for substituting an antecedent with a definition. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  ps )   &    |-  ( ps  ->  ch )   =>    |-  ( ph  ->  ch )
 
Theoremsylib 190 A mixed syllogism inference from an implication and a biconditional. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ps )   &    |-  ( ps 
 <->  ch )   =>    |-  ( ph  ->  ch )
 
Theorembi2 191 Property of the biconditional connective. (Contributed by NM, 11-May-1999.) (Proof shortened by Wolf Lammen, 11-Nov-2012.)
 |-  ( ( ph  <->  ps )  ->  ( ps  ->  ph ) )
 
Theorembicom1 192 Commutative law for equivalence. (Contributed by Wolf Lammen, 10-Nov-2012.)
 |-  ( ( ph  <->  ps )  ->  ( ps 
 <-> 
 ph ) )
 
Theorembicom 193 Commutative law for equivalence. Theorem *4.21 of [WhiteheadRussell] p. 117. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( ph  <->  ps )  <->  ( ps  <->  ph ) )
 
Theorembicomd 194 Commute two sides of a biconditional in a deduction. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( ch  <->  ps ) )
 
Theorembicomi 195 Inference from commutative law for logical equivalence. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  ps )   =>    |-  ( ps  <->  ph )
 
Theoremimpbid1 196 Infer an equivalence from two implications. (Contributed by NM, 6-Mar-2007.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ch  ->  ps )   =>    |-  ( ph  ->  ( ps 
 <->  ch ) )
 
Theoremimpbid2 197 Infer an equivalence from two implications. (Contributed by NM, 6-Mar-2007.) (Proof shortened by Wolf Lammen, 27-Sep-2013.)
 |-  ( ps  ->  ch )   &    |-  ( ph  ->  ( ch  ->  ps ) )   =>    |-  ( ph  ->  ( ps 
 <->  ch ) )
 
Theoremimpcon4bid 198 A variation on impbid 185 with contraposition. (Contributed by Jeff Hankins, 3-Jul-2009.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( -.  ps  ->  -.  ch ) )   =>    |-  ( ph  ->  ( ps  <->  ch ) )
 
Theorembiimpri 199 Infer a converse implication from a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 16-Sep-2013.)
 |-  ( ph  <->  ps )   =>    |-  ( ps  ->  ph )
 
Theorembiimpd 200 Deduce an implication from a logical equivalence. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( ps  ->  ch )
 )
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