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

Theoremloowoz 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.)

1.2.4  Logical conjunction and logical equivalence

Syntaxwa 102 Extend wff definition to include conjunction ('and').

Syntaxwb 103 Extend our wff definition to include the biconditional connective.

Axiomax-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.)

Axiomax-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.)

Axiomax-ia3 106 'And' introduction. One of the axioms of propositional logic. (Contributed by Mario Carneiro, 31-Jan-2015.)

Theoremsimpl 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.)

Theoremsimpr 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.)

Theoremsimpli 109 Inference eliminating a conjunct. (Contributed by NM, 15-Jun-1994.)

Theoremsimpld 110 Deduction eliminating a conjunct. (Contributed by NM, 5-Aug-1993.)

Theoremsimpri 111 Inference eliminating a conjunct. (Contributed by NM, 15-Jun-1994.)

Theoremsimprd 112 Deduction eliminating a conjunct. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)

Theoremex 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.)

Theoremexpcom 114 Exportation inference with commuted antecedents. (Contributed by NM, 25-May-2005.)

Definitiondf-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 921. 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 922) 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 1329, only hold for decidable propositions, not all propositions. (Contributed by NM, 5-Aug-1993.) (Revised by Jim Kingdon, 24-Nov-2017.)

Theorembi1 116 Property of the biconditional connective. (Contributed by NM, 11-May-1999.) (Revised by NM, 31-Jan-2015.)

Theorembi3 117 Property of the biconditional connective. (Contributed by NM, 11-May-1999.)

Theorembiimpi 118 Infer an implication from a logical equivalence. (Contributed by NM, 5-Aug-1993.)

Theoremsylbi 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.)

Theoremsylib 120 A mixed syllogism inference from an implication and a biconditional. (Contributed by NM, 3-Jan-1993.)

Theoremsylbb 121 A mixed syllogism inference from two biconditionals. (Contributed by BJ, 30-Mar-2019.)

Theoremimp 122 Importation inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.)

Theoremimpcom 123 Importation inference with commuted antecedents. (Contributed by NM, 25-May-2005.)

Theoremimpbii 124 Infer an equivalence from an implication and its converse. (Contributed by NM, 5-Aug-1993.)

Theoremimpbidd 125 Deduce an equivalence from two implications. (Contributed by Rodolfo Medina, 12-Oct-2010.)

Theoremimpbid21d 126 Deduce an equivalence from two implications. (Contributed by Wolf Lammen, 12-May-2013.)

Theoremimpbid 127 Deduce an equivalence from two implications. (Contributed by NM, 5-Aug-1993.) (Revised by Wolf Lammen, 3-Nov-2012.)

Theorembi2 128 Property of the biconditional connective. (Contributed by NM, 11-May-1999.) (Proof shortened by Wolf Lammen, 11-Nov-2012.)

Theorembicom1 129 Commutative law for equivalence. (Contributed by Wolf Lammen, 10-Nov-2012.)

Theorembicomi 130 Inference from commutative law for logical equivalence. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 16-Sep-2013.)

Theorembiimpri 131 Infer a converse implication from a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 16-Sep-2013.)

Theoremsylibr 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.)

Theoremsylbir 133 A mixed syllogism inference from a biconditional and an implication. (Contributed by NM, 5-Aug-1993.)

Theoremsylbbr 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.)

Theoremsylbb1 135 A mixed syllogism inference from two biconditionals. (Contributed by BJ, 21-Apr-2019.)

Theoremsylbb2 136 A mixed syllogism inference from two biconditionals. (Contributed by BJ, 21-Apr-2019.)

Theorempm3.2 137 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.)

Theorembicom 138 Commutative law for equivalence. Theorem *4.21 of [WhiteheadRussell] p. 117. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 11-Nov-2012.)

Theorembicomd 139 Commute two sides of a biconditional in a deduction. (Contributed by NM, 5-Aug-1993.)

Theoremimpbid1 140 Infer an equivalence from two implications. (Contributed by NM, 6-Mar-2007.)

Theoremimpbid2 141 Infer an equivalence from two implications. (Contributed by NM, 6-Mar-2007.) (Proof shortened by Wolf Lammen, 27-Sep-2013.)

Theorembiimpd 142 Deduce an implication from a logical equivalence. (Contributed by NM, 5-Aug-1993.)

Theoremmpbi 143 An inference from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.)

Theoremmpbir 144 An inference from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.)

Theoremmpbid 145 A deduction from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.)

Theoremmpbii 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.)

Theoremsylibd 147 A syllogism deduction. (Contributed by NM, 3-Aug-1994.)

Theoremsylbid 148 A syllogism deduction. (Contributed by NM, 3-Aug-1994.)

Theoremmpbidi 149 A deduction from a biconditional, related to modus ponens. (Contributed by NM, 9-Aug-1994.)

Theoremsyl5bi 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.)

Theoremsyl5bir 151 A mixed syllogism inference from a nested implication and a biconditional. (Contributed by NM, 5-Aug-1993.)

Theoremsyl5ib 152 A mixed syllogism inference. (Contributed by NM, 5-Aug-1993.)

Theoremsyl5ibcom 153 A mixed syllogism inference. (Contributed by NM, 19-Jun-2007.)

Theoremsyl5ibr 154 A mixed syllogism inference. (Contributed by NM, 3-Apr-1994.) (Revised by NM, 22-Sep-2013.)

Theoremsyl5ibrcom 155 A mixed syllogism inference. (Contributed by NM, 20-Jun-2007.)

Theorembiimprd 156 Deduce a converse implication from a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 22-Sep-2013.)

Theorembiimpcd 157 Deduce a commuted implication from a logical equivalence. (Contributed by NM, 3-May-1994.) (Proof shortened by Wolf Lammen, 22-Sep-2013.)

Theorembiimprcd 158 Deduce a converse commuted implication from a logical equivalence. (Contributed by NM, 3-May-1994.) (Proof shortened by Wolf Lammen, 20-Dec-2013.)

Theoremsyl6ib 159 A mixed syllogism inference from a nested implication and a biconditional. (Contributed by NM, 5-Aug-1993.)

Theoremsyl6ibr 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.)

Theoremsyl6bi 161 A mixed syllogism inference. (Contributed by NM, 2-Jan-1994.)

Theoremsyl6bir 162 A mixed syllogism inference. (Contributed by NM, 18-May-1994.)

Theoremsyl7bi 163 A mixed syllogism inference from a doubly nested implication and a biconditional. (Contributed by NM, 5-Aug-1993.)

Theoremsyl8ib 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.)

Theoremmpbird 165 A deduction from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.)

Theoremmpbiri 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.)

Theoremsylibrd 167 A syllogism deduction. (Contributed by NM, 3-Aug-1994.)

Theoremsylbird 168 A syllogism deduction. (Contributed by NM, 3-Aug-1994.)

Theorembiid 169 Principle of identity for logical equivalence. Theorem *4.2 of [WhiteheadRussell] p. 117. (Contributed by NM, 5-Aug-1993.)

Theorembiidd 170 Principle of identity with antecedent. (Contributed by NM, 25-Nov-1995.)

Theorempm5.1im 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.)

Theorem2th 172 Two truths are equivalent. (Contributed by NM, 18-Aug-1993.)

Theorem2thd 173 Two truths are equivalent (deduction rule). (Contributed by NM, 3-Jun-2012.) (Revised by NM, 29-Jan-2013.)

Theoremibi 174 Inference that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 17-Oct-2003.)

Theoremibir 175 Inference that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 22-Jul-2004.)

Theoremibd 176 Deduction that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 26-Jun-2004.)

Theorempm5.74 177 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.)

Theorempm5.74i 178 Distribution of implication over biconditional (inference rule). (Contributed by NM, 1-Aug-1994.)

Theorempm5.74ri 179 Distribution of implication over biconditional (reverse inference rule). (Contributed by NM, 1-Aug-1994.)

Theorempm5.74d 180 Distribution of implication over biconditional (deduction rule). (Contributed by NM, 21-Mar-1996.)

Theorempm5.74rd 181 Distribution of implication over biconditional (deduction rule). (Contributed by NM, 19-Mar-1997.)

Theorembitri 182 An inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-Oct-2012.)

Theorembitr2i 183 An inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.)

Theorembitr3i 184 An inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.)

Theorembitr4i 185 An inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.)

Theorembitrd 186 Deduction form of bitri 182. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Apr-2013.)

Theorembitr2d 187 Deduction form of bitr2i 183. (Contributed by NM, 9-Jun-2004.)

Theorembitr3d 188 Deduction form of bitr3i 184. (Contributed by NM, 5-Aug-1993.)

Theorembitr4d 189 Deduction form of bitr4i 185. (Contributed by NM, 5-Aug-1993.)

Theoremsyl5bb 190 A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)

Theoremsyl5rbb 191 A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)

Theoremsyl5bbr 192 A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)

Theoremsyl5rbbr 193 A syllogism inference from two biconditionals. (Contributed by NM, 25-Nov-1994.)

Theoremsyl6bb 194 A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)

Theoremsyl6rbb 195 A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)

Theoremsyl6bbr 196 A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)

Theoremsyl6rbbr 197 A syllogism inference from two biconditionals. (Contributed by NM, 25-Nov-1994.)

Theorem3imtr3i 198 A mixed syllogism inference, useful for removing a definition from both sides of an implication. (Contributed by NM, 10-Aug-1994.)

Theorem3imtr4i 199 A mixed syllogism inference, useful for applying a definition to both sides of an implication. (Contributed by NM, 5-Aug-1993.)

Theorem3imtr3d 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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