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Type | Label | Description |
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Statement | ||
Theorem | loowoz 101 | An alternate for the Linearity Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz, due to Barbara Wozniakowska, Reports on Mathematical Logic 10, 129-137 (1978). (Contributed by O'Cat, 8-Aug-2004.) |
Syntax | wa 102 | Extend wff definition to include conjunction ('and'). |
Syntax | wb 103 | Extend our wff definition to include the biconditional connective. |
Axiom | ax-ia1 104 | Left 'and' elimination. One of the axioms of propositional logic. Use its alias simpl 107 instead for naming consistency with set.mm. (New usage is discouraged.) (Contributed by Mario Carneiro, 31-Jan-2015.) |
Axiom | ax-ia2 105 | Right 'and' elimination. One of the axioms of propositional logic. (Contributed by Mario Carneiro, 31-Jan-2015.) Use its alias simpr 108 instead for naming consistency with set.mm. (New usage is discouraged.) |
Axiom | ax-ia3 106 | 'And' introduction. One of the axioms of propositional logic. (Contributed by Mario Carneiro, 31-Jan-2015.) |
Theorem | simpl 107 | 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 108 | 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 109 | Inference eliminating a conjunct. (Contributed by NM, 15-Jun-1994.) |
Theorem | simpld 110 | Deduction eliminating a conjunct. (Contributed by NM, 5-Aug-1993.) |
Theorem | simpri 111 | Inference eliminating a conjunct. (Contributed by NM, 15-Jun-1994.) |
Theorem | simprd 112 | Deduction eliminating a conjunct. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Oct-2013.) |
Theorem | ex 113 | 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 114 | Exportation inference with commuted antecedents. (Contributed by NM, 25-May-2005.) |
Definition | df-bi 115 |
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 923. 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 924) 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 603. It is impossible to use df-bi 115 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 115 in the proof with the corresponding biijust 603 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 115 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 380 which uses the biconditional instead. Other definitions of the biconditional, such as dfbi3dc 1331, only hold for decidable propositions, not all propositions. (Contributed by NM, 5-Aug-1993.) (Revised by Jim Kingdon, 24-Nov-2017.) |
Theorem | bi1 116 | Property of the biconditional connective. (Contributed by NM, 11-May-1999.) (Revised by NM, 31-Jan-2015.) |
Theorem | bi3 117 | Property of the biconditional connective. (Contributed by NM, 11-May-1999.) |
Theorem | biimpi 118 | Infer an implication from a logical equivalence. (Contributed by NM, 5-Aug-1993.) |
Theorem | sylbi 119 | A mixed syllogism inference from a biconditional and an implication. Useful for substituting an antecedent with a definition. (Contributed by NM, 3-Jan-1993.) |
Theorem | sylib 120 | A mixed syllogism inference from an implication and a biconditional. (Contributed by NM, 3-Jan-1993.) |
Theorem | sylbb 121 | A mixed syllogism inference from two biconditionals. (Contributed by BJ, 30-Mar-2019.) |
Theorem | imp 122 | Importation inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.) |
Theorem | impcom 123 | Importation inference with commuted antecedents. (Contributed by NM, 25-May-2005.) |
Theorem | impbii 124 | Infer an equivalence from an implication and its converse. (Contributed by NM, 5-Aug-1993.) |
Theorem | impbidd 125 | Deduce an equivalence from two implications. (Contributed by Rodolfo Medina, 12-Oct-2010.) |
Theorem | impbid21d 126 | Deduce an equivalence from two implications. (Contributed by Wolf Lammen, 12-May-2013.) |
Theorem | impbid 127 | Deduce an equivalence from two implications. (Contributed by NM, 5-Aug-1993.) (Revised by Wolf Lammen, 3-Nov-2012.) |
Theorem | bi2 128 | Property of the biconditional connective. (Contributed by NM, 11-May-1999.) (Proof shortened by Wolf Lammen, 11-Nov-2012.) |
Theorem | bicom1 129 | Commutative law for equivalence. (Contributed by Wolf Lammen, 10-Nov-2012.) |
Theorem | bicomi 130 | Inference from commutative law for logical equivalence. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 16-Sep-2013.) |
Theorem | biimpri 131 | Infer a converse implication from a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 16-Sep-2013.) |
Theorem | sylibr 132 | 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 | sylbir 133 | A mixed syllogism inference from a biconditional and an implication. (Contributed by NM, 5-Aug-1993.) |
Theorem | sylbbr 134 |
A mixed syllogism inference from two biconditionals.
Note on the various syllogism-like statements in set.mm. The hypothetical syllogism syl 14 infers an implication from two implications (and there are 3syl 17 and 4syl 18 for chaining more inferences). There are four inferences inferring an implication from one implication and one biconditional: sylbi 119, sylib 120, sylbir 133, sylibr 132; four inferences inferring an implication from two biconditionals: sylbb 121, sylbbr 134, sylbb1 135, sylbb2 136; four inferences inferring a biconditional from two biconditionals: bitri 182, bitr2i 183, bitr3i 184, bitr4i 185 (and more for chaining more biconditionals). There are also closed forms and deduction versions of these, like, among many others, syld 44, syl5 32, syl6 33, mpbid 145, bitrd 186, syl5bb 190, syl6bb 194 and variants. (Contributed by BJ, 21-Apr-2019.) |
Theorem | sylbb1 135 | A mixed syllogism inference from two biconditionals. (Contributed by BJ, 21-Apr-2019.) |
Theorem | sylbb2 136 | A mixed syllogism inference from two biconditionals. (Contributed by BJ, 21-Apr-2019.) |
Theorem | pm3.2 137 | 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 | bicom 138 | 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 139 | Commute two sides of a biconditional in a deduction. (Contributed by NM, 5-Aug-1993.) |
Theorem | impbid1 140 | Infer an equivalence from two implications. (Contributed by NM, 6-Mar-2007.) |
Theorem | impbid2 141 | Infer an equivalence from two implications. (Contributed by NM, 6-Mar-2007.) (Proof shortened by Wolf Lammen, 27-Sep-2013.) |
Theorem | biimpd 142 | Deduce an implication from a logical equivalence. (Contributed by NM, 5-Aug-1993.) |
Theorem | mpbi 143 | An inference from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.) |
Theorem | mpbir 144 | An inference from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.) |
Theorem | mpbid 145 | A deduction from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.) |
Theorem | mpbii 146 | 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 | sylibd 147 | A syllogism deduction. (Contributed by NM, 3-Aug-1994.) |
Theorem | sylbid 148 | A syllogism deduction. (Contributed by NM, 3-Aug-1994.) |
Theorem | mpbidi 149 | A deduction from a biconditional, related to modus ponens. (Contributed by NM, 9-Aug-1994.) |
Theorem | syl5bi 150 | 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 151 | A mixed syllogism inference from a nested implication and a biconditional. (Contributed by NM, 5-Aug-1993.) |
Theorem | syl5ib 152 | A mixed syllogism inference. (Contributed by NM, 5-Aug-1993.) |
Theorem | syl5ibcom 153 | A mixed syllogism inference. (Contributed by NM, 19-Jun-2007.) |
Theorem | syl5ibr 154 | A mixed syllogism inference. (Contributed by NM, 3-Apr-1994.) (Revised by NM, 22-Sep-2013.) |
Theorem | syl5ibrcom 155 | A mixed syllogism inference. (Contributed by NM, 20-Jun-2007.) |
Theorem | biimprd 156 | Deduce a converse implication from a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 22-Sep-2013.) |
Theorem | biimpcd 157 | Deduce a commuted implication from a logical equivalence. (Contributed by NM, 3-May-1994.) (Proof shortened by Wolf Lammen, 22-Sep-2013.) |
Theorem | biimprcd 158 | 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 159 | A mixed syllogism inference from a nested implication and a biconditional. (Contributed by NM, 5-Aug-1993.) |
Theorem | syl6ibr 160 | 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 161 | A mixed syllogism inference. (Contributed by NM, 2-Jan-1994.) |
Theorem | syl6bir 162 | A mixed syllogism inference. (Contributed by NM, 18-May-1994.) |
Theorem | syl7bi 163 | A mixed syllogism inference from a doubly nested implication and a biconditional. (Contributed by NM, 5-Aug-1993.) |
Theorem | syl8ib 164 | 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 165 | A deduction from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.) |
Theorem | mpbiri 166 | 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 167 | A syllogism deduction. (Contributed by NM, 3-Aug-1994.) |
Theorem | sylbird 168 | A syllogism deduction. (Contributed by NM, 3-Aug-1994.) |
Theorem | biid 169 | Principle of identity for logical equivalence. Theorem *4.2 of [WhiteheadRussell] p. 117. (Contributed by NM, 5-Aug-1993.) |
Theorem | biidd 170 | Principle of identity with antecedent. (Contributed by NM, 25-Nov-1995.) |
Theorem | pm5.1im 171 | Two propositions are equivalent if they are both true. Closed form of 2th 172. Equivalent to a bi1 116-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 172 | Two truths are equivalent. (Contributed by NM, 18-Aug-1993.) |
Theorem | 2thd 173 | Two truths are equivalent (deduction rule). (Contributed by NM, 3-Jun-2012.) (Revised by NM, 29-Jan-2013.) |
Theorem | ibi 174 | Inference that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 17-Oct-2003.) |
Theorem | ibir 175 | Inference that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 22-Jul-2004.) |
Theorem | ibd 176 | Deduction that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 26-Jun-2004.) |
Theorem | pm5.74 177 | 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 178 | Distribution of implication over biconditional (inference rule). (Contributed by NM, 1-Aug-1994.) |
Theorem | pm5.74ri 179 | Distribution of implication over biconditional (reverse inference rule). (Contributed by NM, 1-Aug-1994.) |
Theorem | pm5.74d 180 | Distribution of implication over biconditional (deduction rule). (Contributed by NM, 21-Mar-1996.) |
Theorem | pm5.74rd 181 | Distribution of implication over biconditional (deduction rule). (Contributed by NM, 19-Mar-1997.) |
Theorem | bitri 182 | An inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-Oct-2012.) |
Theorem | bitr2i 183 | An inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.) |
Theorem | bitr3i 184 | An inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.) |
Theorem | bitr4i 185 | An inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.) |
Theorem | bitrd 186 | Deduction form of bitri 182. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Apr-2013.) |
Theorem | bitr2d 187 | Deduction form of bitr2i 183. (Contributed by NM, 9-Jun-2004.) |
Theorem | bitr3d 188 | Deduction form of bitr3i 184. (Contributed by NM, 5-Aug-1993.) |
Theorem | bitr4d 189 | Deduction form of bitr4i 185. (Contributed by NM, 5-Aug-1993.) |
Theorem | syl5bb 190 | A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.) |
Theorem | syl5rbb 191 | A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.) |
Theorem | syl5bbr 192 | A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.) |
Theorem | syl5rbbr 193 | A syllogism inference from two biconditionals. (Contributed by NM, 25-Nov-1994.) |
Theorem | syl6bb 194 | A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.) |
Theorem | syl6rbb 195 | A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.) |
Theorem | syl6bbr 196 | A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.) |
Theorem | syl6rbbr 197 | A syllogism inference from two biconditionals. (Contributed by NM, 25-Nov-1994.) |
Theorem | 3imtr3i 198 | A mixed syllogism inference, useful for removing a definition from both sides of an implication. (Contributed by NM, 10-Aug-1994.) |
Theorem | 3imtr4i 199 | A mixed syllogism inference, useful for applying a definition to both sides of an implication. (Contributed by NM, 5-Aug-1993.) |
Theorem | 3imtr3d 200 | More general version of 3imtr3i 198. Useful for converting conditional definitions in a formula. (Contributed by NM, 8-Apr-1996.) |
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