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Theorem List for Intuitionistic Logic Explorer - 101-200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsimpl 101 Elimination of a conjunct. Theorem *3.26 (Simp) of [WhiteheadRussell] p. 112. (The proof was shortened by Wolf Lammen, 13-Nov-2012.)
 
Theoremsimpr 102 Elimination of a conjunct. Theorem *3.27 (Simp) of [WhiteheadRussell] p. 112. (The proof was shortened by Wolf Lammen, 13-Nov-2012.)
 
Theoremsimpli 103 Inference eliminating a conjunct.
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Theoremsimpld 104 Deduction eliminating a conjunct.
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Theoremsimpri 105 Inference eliminating a conjunct.
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Theoremsimprd 106 Deduction eliminating a conjunct. (The proof was shortened by Wolf Lammen, 3-Oct-2013.)
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Theoremex 107 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.) (The proof was shortened by Eric Schmidt, 22-Dec-2006.)
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Theoremexpcom 108 Exportation inference with commuted antecedents.
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Definitiondf-bi 109 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 (df-or 789 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 905) 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 bijust 549. It is impossible to use df-bi 109 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 109 in the proof with the corresponding bijust 549 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.)

See dfbi1 778, dfbi2 366, and dfbi3 825 for theorems suggesting typical textbook definitions of , showing that our definition has the properties we expect. Theorem dfbi 367 shows this definition rewritten in an abbreviated form after conjunction is introduced, for easier understanding.

 
Theorembi1 110 Property of the biconditional connective.
 
Theorembi3 111 Property of the biconditional connective.
 
Theorembiimpi 112 Infer an implication from a logical equivalence.
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Theoremsylbi 113 A mixed syllogism inference from a biconditional and an implication. Useful for substituting an antecedent with a definition.
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Theoremimp 114 Importation inference. (The proof was shortened by Eric Schmidt, 22-Dec-2006.)
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Theoremimpcom 115 Importation inference with commuted antecedents.
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Theoremimpbii 116 Infer an equivalence from an implication and its converse.
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Theoremimpbidd 117 Deduce an equivalence from two implications. (Contributed by Rodolfo Medina, 12-Oct-2010.)
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Theoremimpbid21d 118 Deduce an equivalence from two implications. (Contributed by Wolf Lammen, 12-May-2013.)
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Theoremimpbid 119 Deduce an equivalence from two implications. (Dependency on df-an removed by Wolf Lammen, 3-Nov-2012.)
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Theorembi2 120 Property of the biconditional connective. (The proof was shortened by Wolf Lammen, 11-Nov-2012.)
 
Theorembicom1 121 Commutative law for equivalence. (Contributed by Wolf Lammen, 10-Nov-2012.)
 
Theorembicomi 122 Inference from commutative law for logical equivalence.
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Theorembiimpri 123 Infer a converse implication from a logical equivalence. (The proof was shortened by Wolf Lammen, 16-Sep-2013.)
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Theoremsylbir 124 A mixed syllogism inference from a biconditional and an implication.
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Theorempm3.2 125 Join antecedents with conjunction. Theorem *3.2 of [WhiteheadRussell] p. 111. (The proof was shortened by Wolf Lammen, 12-Nov-2012.)
 
Theoremsylib 126 A mixed syllogism inference from an implication and a biconditional.
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Theorembicom 127 Commutative law for equivalence. Theorem *4.21 of [WhiteheadRussell] p. 117.
 
Theorembicomd 128 Commute two sides of a biconditional in a deduction.
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Theoremimpbid1 129 Infer an equivalence from two implications.
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Theoremimpbid2 130 Infer an equivalence from two implications. (The proof was shortened by Wolf Lammen, 27-Sep-2013.)
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Theorembiimpd 131 Deduce an implication from a logical equivalence.
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Theoremmpbi 132 An inference from a biconditional, related to modus ponens.
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Theoremmpbir 133 An inference from a biconditional, related to modus ponens.
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Theoremmpbid 134 A deduction from a biconditional, related to modus ponens.
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Theoremmpbii 135 An inference from a nested biconditional, related to modus ponens. (The proof was shortened by Wolf Lammen, 25-Oct-2012.)
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Theoremsylibr 136 A mixed syllogism inference from an implication and a biconditional. Useful for substituting a consequent with a definition.
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Theoremsylibd 137 A syllogism deduction.
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Theoremsylbid 138 A syllogism deduction.
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Theoremmpbidi 139 A deduction from a biconditional, related to modus ponens.
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Theoremsyl5bi 140 A mixed syllogism inference from a nested implication and a biconditional. Useful for substituting an embedded antecedent with a definition.
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Theoremsyl5bir 141 A mixed syllogism inference from a nested implication and a biconditional.
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Theoremsyl5ib 142 A mixed syllogism inference.
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Theoremsyl5ibcom 143 A mixed syllogism inference.
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Theoremsyl5ibr 144 A mixed syllogism inference.
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Theoremsyl5ibrcom 145 A mixed syllogism inference.
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Theorembiimprd 146 Deduce a converse implication from a logical equivalence. (The proof was shortened by Wolf Lammen, 22-Sep-2013.)
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Theorembiimpcd 147 Deduce a commuted implication from a logical equivalence. (The proof was shortened by Wolf Lammen, 22-Sep-2013.)
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Theorembiimprcd 148 Deduce a converse commuted implication from a logical equivalence. (The proof was shortened by Wolf Lammen, 20-Dec-2013.)
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Theoremsyl6ib 149 A mixed syllogism inference from a nested implication and a biconditional.
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Theoremsyl6ibr 150 A mixed syllogism inference from a nested implication and a biconditional. Useful for substituting an embedded consequent with a definition.
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Theoremsyl6bi 151 A mixed syllogism inference.
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Theoremsyl6bir 152 A mixed syllogism inference.
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Theoremsyl7bi 153 A mixed syllogism inference from a doubly nested implication and a biconditional.
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Theoremsyl8ib 154 A syllogism rule of inference. The second premise is used to replace the consequent of the first premise.
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Theoremmpbird 155 A deduction from a biconditional, related to modus ponens.
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Theoremmpbiri 156 An inference from a nested biconditional, related to modus ponens. (The proof was shortened by Wolf Lammen, 25-Oct-2012.)
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Theoremsylibrd 157 A syllogism deduction.
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Theoremsylbird 158 A syllogism deduction.
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Theorembiid 159 Principle of identity for logical equivalence. Theorem *4.2 of [WhiteheadRussell] p. 117.
 
Theorembiidd 160 Principle of identity with antecedent.
 
Theorempm5.1im 161 Two propositions are equivalent if they are both true. Closed form of 2th 162. Equivalent to a bi1 110-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 162 Two truths are equivalent.
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Theorem2thd 163 Two truths are equivalent (deduction rule).
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Theoremibi 164 Inference that converts a biconditional implied by one of its arguments, into an implication.
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Theoremibir 165 Inference that converts a biconditional implied by one of its arguments, into an implication.
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Theoremibd 166 Deduction that converts a biconditional implied by one of its arguments, into an implication.
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Theorempm5.74 167 Distribution of implication over biconditional. Theorem *5.74 of [WhiteheadRussell] p. 126. (The proof was shortened by Wolf Lammen, 11-Apr-2013.)
 
Theorempm5.74i 168 Distribution of implication over biconditional (inference rule).
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Theorempm5.74ri 169 Distribution of implication over biconditional (reverse inference rule).
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Theorempm5.74d 170 Distribution of implication over biconditional (deduction rule).
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Theorempm5.74rd 171 Distribution of implication over biconditional (deduction rule).
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Theorembitri 172 An inference from transitive law for logical equivalence. (The proof was shortened by Wolf Lammen, 13-Oct-2012.)
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Theorembitr2i 173 An inference from transitive law for logical equivalence.
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Theorembitr3i 174 An inference from transitive law for logical equivalence.
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Theorembitr4i 175 An inference from transitive law for logical equivalence.
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Theorembitrd 176 Deduction form of bitri 172. (The proof was shortened by Wolf Lammen, 14-Apr-2013.)
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Theorembitr2d 177 Deduction form of bitr2i 173.
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Theorembitr3d 178 Deduction form of bitr3i 174.
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Theorembitr4d 179 Deduction form of bitr4i 175.
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Theoremsyl5bb 180 A syllogism inference from two biconditionals.
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Theoremsyl5rbb 181 A syllogism inference from two biconditionals.
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Theoremsyl5bbr 182 A syllogism inference from two biconditionals.
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Theoremsyl5rbbr 183 A syllogism inference from two biconditionals.
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Theoremsyl6bb 184 A syllogism inference from two biconditionals.
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Theoremsyl6rbb 185 A syllogism inference from two biconditionals.
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Theoremsyl6bbr 186 A syllogism inference from two biconditionals.
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Theoremsyl6rbbr 187 A syllogism inference from two biconditionals.
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Theorem3imtr3i 188 A mixed syllogism inference, useful for removing a definition from both sides of an implication.
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Theorem3imtr4i 189 A mixed syllogism inference, useful for applying a definition to both sides of an implication.
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Theorem3imtr3d 190 More general version of 3imtr3i 188. Useful for converting conditional definitions in a formula.
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Theorem3imtr4d 191 More general version of 3imtr4i 189. Useful for converting conditional definitions in a formula.
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Theorem3imtr3g 192 More general version of 3imtr3i 188. Useful for converting definitions in a formula. (The proof was shortened by Wolf Lammen, 20-Dec-2013.)
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Theorem3imtr4g 193 More general version of 3imtr4i 189. Useful for converting definitions in a formula. (The proof was shortened by Wolf Lammen, 20-Dec-2013.)
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Theorem3bitri 194 A chained inference from transitive law for logical equivalence.
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Theorem3bitrri 195 A chained inference from transitive law for logical equivalence.
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Theorem3bitr2i 196 A chained inference from transitive law for logical equivalence.
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Theorem3bitr2ri 197 A chained inference from transitive law for logical equivalence.
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Theorem3bitr3i 198 A chained inference from transitive law for logical equivalence.
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Theorem3bitr3ri 199 A chained inference from transitive law for logical equivalence.
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Theorem3bitr4i 200 A chained inference from transitive law for logical equivalence. This inference is frequently used to apply a definition to both sides of a logical equivalence.
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