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Type | Label | Description |
---|---|---|
Statement | ||
Syntax | wa 101 | Extend wff definition to include conjunction ('and'). |
Syntax | wb 102 | Extend our wff definition to include the biconditional connective. |
Axiom | ax-ia1 103 | Left 'and' elimination. One of the axioms of propositional logic. Use its alias simpl 106 instead for naming consistency with set.mm. (New usage is discouraged.) (Contributed by Mario Carneiro, 31-Jan-2015.) |
Axiom | ax-ia2 104 | Right 'and' elimination. One of the axioms of propositional logic. (Contributed by Mario Carneiro, 31-Jan-2015.) Use its alias simpr 107 instead for naming consistency with set.mm. (New usage is discouraged.) |
Axiom | ax-ia3 105 | 'And' introduction. One of the axioms of propositional logic. (Contributed by Mario Carneiro, 31-Jan-2015.) |
Theorem | simpl 106 | Elimination of a conjunct. Theorem *3.26 (Simp) of [WhiteheadRussell] p. 112. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-Nov-2012.) |
Theorem | simpr 107 | Elimination of a conjunct. Theorem *3.27 (Simp) of [WhiteheadRussell] p. 112. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-Nov-2012.) |
Theorem | simpli 108 | Inference eliminating a conjunct. (Contributed by NM, 15-Jun-1994.) |
Theorem | simpld 109 | Deduction eliminating a conjunct. (Contributed by NM, 5-Aug-1993.) |
Theorem | simpri 110 | Inference eliminating a conjunct. (Contributed by NM, 15-Jun-1994.) |
Theorem | simprd 111 | Deduction eliminating a conjunct. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Oct-2013.) |
Theorem | ex 112 | Exportation inference. (This theorem used to be labeled "exp" but was changed to "ex" so as not to conflict with the math token "exp", per the June 2006 Metamath spec change.) (Contributed by NM, 5-Aug-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.) |
Theorem | expcom 113 | Exportation inference with commuted antecedents. (Contributed by NM, 25-May-2005.) |
Definition | df-bi 114 |
This is our first definition, which introduces and defines the
biconditional connective . We define a wff of the form
as an abbreviation for
.
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, as it allows us to use logic to manipulate definitions directly. For an example of such a definition, see df-3or 897. 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 898) 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 (the definiendum i.e. the thing being defined) with (the definiens i.e. the defining expression) in the definition, the definition becomes the previously proved theorem biijust 580. It is impossible to use df-bi 114 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 114 in the proof with the corresponding biijust 580 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 for human readability.) df-bi 114 itself is a conjunction of two implications (to avoid using the biconditional in its own definition), but once we have the biconditional, we can prove dfbi2 374 which uses the biconditional instead. Other definitions of the biconditional, such as dfbi3dc 1304, only hold for decidable propositions, not all propositions. (Contributed by NM, 5-Aug-1993.) (Revised by Jim Kingdon, 24-Nov-2017.) |
Theorem | bi1 115 | Property of the biconditional connective. (Contributed by NM, 11-May-1999.) (Revised by NM, 31-Jan-2015.) |
Theorem | bi3 116 | Property of the biconditional connective. (Contributed by NM, 11-May-1999.) |
Theorem | biimpi 117 | Infer an implication from a logical equivalence. (Contributed by NM, 5-Aug-1993.) |
Theorem | sylbi 118 | A mixed syllogism inference from a biconditional and an implication. Useful for substituting an antecedent with a definition. (Contributed by NM, 5-Aug-1993.) |
Theorem | imp 119 | Importation inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.) |
Theorem | impcom 120 | Importation inference with commuted antecedents. (Contributed by NM, 25-May-2005.) |
Theorem | impbii 121 | Infer an equivalence from an implication and its converse. (Contributed by NM, 5-Aug-1993.) |
Theorem | impbidd 122 | Deduce an equivalence from two implications. (Contributed by Rodolfo Medina, 12-Oct-2010.) |
Theorem | impbid21d 123 | Deduce an equivalence from two implications. (Contributed by Wolf Lammen, 12-May-2013.) |
Theorem | impbid 124 | Deduce an equivalence from two implications. (Contributed by NM, 5-Aug-1993.) (Revised by Wolf Lammen, 3-Nov-2012.) |
Theorem | bi2 125 | Property of the biconditional connective. (Contributed by NM, 11-May-1999.) (Proof shortened by Wolf Lammen, 11-Nov-2012.) |
Theorem | bicom1 126 | Commutative law for equivalence. (Contributed by Wolf Lammen, 10-Nov-2012.) |
Theorem | bicomi 127 | Inference from commutative law for logical equivalence. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 16-Sep-2013.) |
Theorem | biimpri 128 | Infer a converse implication from a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 16-Sep-2013.) |
Theorem | sylbir 129 | A mixed syllogism inference from a biconditional and an implication. (Contributed by NM, 5-Aug-1993.) |
Theorem | pm3.2 130 | Join antecedents with conjunction. Theorem *3.2 of [WhiteheadRussell] p. 111. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Nov-2012.) (Proof shortened by Jia Ming, 17-Nov-2020.) |
Theorem | sylib 131 | A mixed syllogism inference from an implication and a biconditional. (Contributed by NM, 5-Aug-1993.) |
Theorem | bicom 132 | Commutative law for equivalence. Theorem *4.21 of [WhiteheadRussell] p. 117. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 11-Nov-2012.) |
Theorem | bicomd 133 | Commute two sides of a biconditional in a deduction. (Contributed by NM, 5-Aug-1993.) |
Theorem | impbid1 134 | Infer an equivalence from two implications. (Contributed by NM, 6-Mar-2007.) |
Theorem | impbid2 135 | Infer an equivalence from two implications. (Contributed by NM, 6-Mar-2007.) (Proof shortened by Wolf Lammen, 27-Sep-2013.) |
Theorem | biimpd 136 | Deduce an implication from a logical equivalence. (Contributed by NM, 5-Aug-1993.) |
Theorem | mpbi 137 | An inference from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.) |
Theorem | mpbir 138 | An inference from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.) |
Theorem | mpbid 139 | A deduction from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.) |
Theorem | mpbii 140 | An inference from a nested biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2012.) |
Theorem | sylibr 141 | A mixed syllogism inference from an implication and a biconditional. Useful for substituting a consequent with a definition. (Contributed by NM, 5-Aug-1993.) |
Theorem | sylibd 142 | A syllogism deduction. (Contributed by NM, 3-Aug-1994.) |
Theorem | sylbid 143 | A syllogism deduction. (Contributed by NM, 3-Aug-1994.) |
Theorem | mpbidi 144 | A deduction from a biconditional, related to modus ponens. (Contributed by NM, 9-Aug-1994.) |
Theorem | syl5bi 145 | A mixed syllogism inference from a nested implication and a biconditional. Useful for substituting an embedded antecedent with a definition. (Contributed by NM, 5-Aug-1993.) |
Theorem | syl5bir 146 | A mixed syllogism inference from a nested implication and a biconditional. (Contributed by NM, 5-Aug-1993.) |
Theorem | syl5ib 147 | A mixed syllogism inference. (Contributed by NM, 5-Aug-1993.) |
Theorem | syl5ibcom 148 | A mixed syllogism inference. (Contributed by NM, 19-Jun-2007.) |
Theorem | syl5ibr 149 | A mixed syllogism inference. (Contributed by NM, 3-Apr-1994.) (Revised by NM, 22-Sep-2013.) |
Theorem | syl5ibrcom 150 | A mixed syllogism inference. (Contributed by NM, 20-Jun-2007.) |
Theorem | biimprd 151 | Deduce a converse implication from a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 22-Sep-2013.) |
Theorem | biimpcd 152 | Deduce a commuted implication from a logical equivalence. (Contributed by NM, 3-May-1994.) (Proof shortened by Wolf Lammen, 22-Sep-2013.) |
Theorem | biimprcd 153 | Deduce a converse commuted implication from a logical equivalence. (Contributed by NM, 3-May-1994.) (Proof shortened by Wolf Lammen, 20-Dec-2013.) |
Theorem | syl6ib 154 | A mixed syllogism inference from a nested implication and a biconditional. (Contributed by NM, 5-Aug-1993.) |
Theorem | syl6ibr 155 | A mixed syllogism inference from a nested implication and a biconditional. Useful for substituting an embedded consequent with a definition. (Contributed by NM, 5-Aug-1993.) |
Theorem | syl6bi 156 | A mixed syllogism inference. (Contributed by NM, 2-Jan-1994.) |
Theorem | syl6bir 157 | A mixed syllogism inference. (Contributed by NM, 18-May-1994.) |
Theorem | syl7bi 158 | A mixed syllogism inference from a doubly nested implication and a biconditional. (Contributed by NM, 5-Aug-1993.) |
Theorem | syl8ib 159 | A syllogism rule of inference. The second premise is used to replace the consequent of the first premise. (Contributed by NM, 1-Aug-1994.) |
Theorem | mpbird 160 | A deduction from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.) |
Theorem | mpbiri 161 | An inference from a nested biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2012.) |
Theorem | sylibrd 162 | A syllogism deduction. (Contributed by NM, 3-Aug-1994.) |
Theorem | sylbird 163 | A syllogism deduction. (Contributed by NM, 3-Aug-1994.) |
Theorem | biid 164 | Principle of identity for logical equivalence. Theorem *4.2 of [WhiteheadRussell] p. 117. (Contributed by NM, 5-Aug-1993.) |
Theorem | biidd 165 | Principle of identity with antecedent. (Contributed by NM, 25-Nov-1995.) |
Theorem | pm5.1im 166 | Two propositions are equivalent if they are both true. Closed form of 2th 167. Equivalent to a bi1 115-like version of the xor-connective. This theorem stays true, no matter how you permute its operands. This is evident from its sharper version . (Contributed by Wolf Lammen, 12-May-2013.) |
Theorem | 2th 167 | Two truths are equivalent. (Contributed by NM, 18-Aug-1993.) |
Theorem | 2thd 168 | Two truths are equivalent (deduction rule). (Contributed by NM, 3-Jun-2012.) (Revised by NM, 29-Jan-2013.) |
Theorem | ibi 169 | Inference that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 17-Oct-2003.) |
Theorem | ibir 170 | Inference that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 22-Jul-2004.) |
Theorem | ibd 171 | Deduction that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 26-Jun-2004.) |
Theorem | pm5.74 172 | Distribution of implication over biconditional. Theorem *5.74 of [WhiteheadRussell] p. 126. (Contributed by NM, 1-Aug-1994.) (Proof shortened by Wolf Lammen, 11-Apr-2013.) |
Theorem | pm5.74i 173 | Distribution of implication over biconditional (inference rule). (Contributed by NM, 1-Aug-1994.) |
Theorem | pm5.74ri 174 | Distribution of implication over biconditional (reverse inference rule). (Contributed by NM, 1-Aug-1994.) |
Theorem | pm5.74d 175 | Distribution of implication over biconditional (deduction rule). (Contributed by NM, 21-Mar-1996.) |
Theorem | pm5.74rd 176 | Distribution of implication over biconditional (deduction rule). (Contributed by NM, 19-Mar-1997.) |
Theorem | bitri 177 | An inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-Oct-2012.) |
Theorem | bitr2i 178 | An inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.) |
Theorem | bitr3i 179 | An inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.) |
Theorem | bitr4i 180 | An inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.) |
Theorem | bitrd 181 | Deduction form of bitri 177. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Apr-2013.) |
Theorem | bitr2d 182 | Deduction form of bitr2i 178. (Contributed by NM, 9-Jun-2004.) |
Theorem | bitr3d 183 | Deduction form of bitr3i 179. (Contributed by NM, 5-Aug-1993.) |
Theorem | bitr4d 184 | Deduction form of bitr4i 180. (Contributed by NM, 5-Aug-1993.) |
Theorem | syl5bb 185 | A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.) |
Theorem | syl5rbb 186 | A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.) |
Theorem | syl5bbr 187 | A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.) |
Theorem | syl5rbbr 188 | A syllogism inference from two biconditionals. (Contributed by NM, 25-Nov-1994.) |
Theorem | syl6bb 189 | A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.) |
Theorem | syl6rbb 190 | A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.) |
Theorem | syl6bbr 191 | A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.) |
Theorem | syl6rbbr 192 | A syllogism inference from two biconditionals. (Contributed by NM, 25-Nov-1994.) |
Theorem | 3imtr3i 193 | A mixed syllogism inference, useful for removing a definition from both sides of an implication. (Contributed by NM, 10-Aug-1994.) |
Theorem | 3imtr4i 194 | A mixed syllogism inference, useful for applying a definition to both sides of an implication. (Contributed by NM, 5-Aug-1993.) |
Theorem | 3imtr3d 195 | More general version of 3imtr3i 193. Useful for converting conditional definitions in a formula. (Contributed by NM, 8-Apr-1996.) |
Theorem | 3imtr4d 196 | More general version of 3imtr4i 194. Useful for converting conditional definitions in a formula. (Contributed by NM, 26-Oct-1995.) |
Theorem | 3imtr3g 197 | More general version of 3imtr3i 193. Useful for converting definitions in a formula. (Contributed by NM, 20-May-1996.) (Proof shortened by Wolf Lammen, 20-Dec-2013.) |
Theorem | 3imtr4g 198 | More general version of 3imtr4i 194. Useful for converting definitions in a formula. (Contributed by NM, 20-May-1996.) (Proof shortened by Wolf Lammen, 20-Dec-2013.) |
Theorem | 3bitri 199 | A chained inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.) |
Theorem | 3bitrri 200 | A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.) |
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