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Theorem List for Intuitionistic Logic Explorer - 101-200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsimpr 101 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 102 Inference eliminating a conjunct. (Contributed by NM, 15-Jun-1994.)
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Theoremsimpld 103 Deduction eliminating a conjunct. (Contributed by NM, 5-Aug-1993.)
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Theoremsimpri 104 Inference eliminating a conjunct. (Contributed by NM, 15-Jun-1994.)
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Theoremsimprd 105 Deduction eliminating a conjunct. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
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Theoremex 106 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.)
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Theoremexpcom 107 Exportation inference with commuted antecedents. (Contributed by NM, 25-May-2005.)
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Definitiondf-bi 108 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 844. 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 845) 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 551. It is impossible to use df-bi 108 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 108 in the proof with the corresponding biijust 551 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 108 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 365 which uses the biconditional instead.

Other definitions of the biconditional, such as dfbi3dc 1222, only hold for decidable propositions, not all propositions. (Contributed by NM, 5-Aug-1993.) (Revised by Jim Kingdon, 24-Nov-2017.)

 
Theorembi1 109 Property of the biconditional connective. (Contributed by NM, 11-May-1999.) (Revised by NM, 31-Jan-2015.)
 
Theorembi3 110 Property of the biconditional connective. (Contributed by NM, 11-May-1999.)
 
Theorembiimpi 111 Infer an implication from a logical equivalence. (Contributed by NM, 5-Aug-1993.)
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Theoremsylbi 112 A mixed syllogism inference from a biconditional and an implication. Useful for substituting an antecedent with a definition. (Contributed by NM, 5-Aug-1993.)
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Theoremimp 113 Importation inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.)
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Theoremimpcom 114 Importation inference with commuted antecedents. (Contributed by NM, 25-May-2005.)
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Theoremimpbii 115 Infer an equivalence from an implication and its converse. (Contributed by NM, 5-Aug-1993.)
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Theoremimpbidd 116 Deduce an equivalence from two implications. (Contributed by Rodolfo Medina, 12-Oct-2010.)
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Theoremimpbid21d 117 Deduce an equivalence from two implications. (Contributed by Wolf Lammen, 12-May-2013.)
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Theoremimpbid 118 Deduce an equivalence from two implications. (Contributed by NM, 5-Aug-1993.) (Revised by Wolf Lammen, 3-Nov-2012.)
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Theorembi2 119 Property of the biconditional connective. (Contributed by NM, 11-May-1999.) (Proof shortened by Wolf Lammen, 11-Nov-2012.)
 
Theorembicom1 120 Commutative law for equivalence. (Contributed by Wolf Lammen, 10-Nov-2012.)
 
Theorembicomi 121 Inference from commutative law for logical equivalence. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 16-Sep-2013.)
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Theorembiimpri 122 Infer a converse implication from a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 16-Sep-2013.)
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Theoremsylbir 123 A mixed syllogism inference from a biconditional and an implication. (Contributed by NM, 5-Aug-1993.)
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Theorempm3.2 124 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.)
 
Theoremsylib 125 A mixed syllogism inference from an implication and a biconditional. (Contributed by NM, 5-Aug-1993.)
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Theorembicom 126 Commutative law for equivalence. Theorem *4.21 of [WhiteheadRussell] p. 117. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 11-Nov-2012.)
 
Theorembicomd 127 Commute two sides of a biconditional in a deduction. (Contributed by NM, 5-Aug-1993.)
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Theoremimpbid1 128 Infer an equivalence from two implications. (Contributed by NM, 6-Mar-2007.)
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Theoremimpbid2 129 Infer an equivalence from two implications. (Contributed by NM, 6-Mar-2007.) (Proof shortened by Wolf Lammen, 27-Sep-2013.)
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Theorembiimpd 130 Deduce an implication from a logical equivalence. (Contributed by NM, 5-Aug-1993.)
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Theoremmpbi 131 An inference from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.)
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Theoremmpbir 132 An inference from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.)
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Theoremmpbid 133 A deduction from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.)
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Theoremmpbii 134 An inference from a nested biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2012.)
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Theoremsylibr 135 A mixed syllogism inference from an implication and a biconditional. Useful for substituting a consequent with a definition. (Contributed by NM, 5-Aug-1993.)
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Theoremsylibd 136 A syllogism deduction. (Contributed by NM, 3-Aug-1994.)
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Theoremsylbid 137 A syllogism deduction. (Contributed by NM, 3-Aug-1994.)
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Theoremmpbidi 138 A deduction from a biconditional, related to modus ponens. (Contributed by NM, 9-Aug-1994.)
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Theoremsyl5bi 139 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.)
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Theoremsyl5bir 140 A mixed syllogism inference from a nested implication and a biconditional. (Contributed by NM, 5-Aug-1993.)
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Theoremsyl5ib 141 A mixed syllogism inference. (Contributed by NM, 5-Aug-1993.)
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Theoremsyl5ibcom 142 A mixed syllogism inference. (Contributed by NM, 19-Jun-2007.)
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Theoremsyl5ibr 143 A mixed syllogism inference. (Contributed by NM, 3-Apr-1994.) (Revised by NM, 22-Sep-2013.)
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Theoremsyl5ibrcom 144 A mixed syllogism inference. (Contributed by NM, 20-Jun-2007.)
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Theorembiimprd 145 Deduce a converse implication from a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 22-Sep-2013.)
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Theorembiimpcd 146 Deduce a commuted implication from a logical equivalence. (Contributed by NM, 3-May-1994.) (Proof shortened by Wolf Lammen, 22-Sep-2013.)
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Theorembiimprcd 147 Deduce a converse commuted implication from a logical equivalence. (Contributed by NM, 3-May-1994.) (Proof shortened by Wolf Lammen, 20-Dec-2013.)
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Theoremsyl6ib 148 A mixed syllogism inference from a nested implication and a biconditional. (Contributed by NM, 5-Aug-1993.)
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Theoremsyl6ibr 149 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.)
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Theoremsyl6bi 150 A mixed syllogism inference. (Contributed by NM, 2-Jan-1994.)
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Theoremsyl6bir 151 A mixed syllogism inference. (Contributed by NM, 18-May-1994.)
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Theoremsyl7bi 152 A mixed syllogism inference from a doubly nested implication and a biconditional. (Contributed by NM, 5-Aug-1993.)
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Theoremsyl8ib 153 A syllogism rule of inference. The second premise is used to replace the consequent of the first premise. (Contributed by NM, 1-Aug-1994.)
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Theoremmpbird 154 A deduction from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.)
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Theoremmpbiri 155 An inference from a nested biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2012.)
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Theoremsylibrd 156 A syllogism deduction. (Contributed by NM, 3-Aug-1994.)
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Theoremsylbird 157 A syllogism deduction. (Contributed by NM, 3-Aug-1994.)
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Theorembiid 158 Principle of identity for logical equivalence. Theorem *4.2 of [WhiteheadRussell] p. 117. (Contributed by NM, 5-Aug-1993.)
 
Theorembiidd 159 Principle of identity with antecedent. (Contributed by NM, 25-Nov-1995.)
 
Theorempm5.1im 160 Two propositions are equivalent if they are both true. Closed form of 2th 161. Equivalent to a bi1 109-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 161 Two truths are equivalent. (Contributed by NM, 18-Aug-1993.)
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Theorem2thd 162 Two truths are equivalent (deduction rule). (Contributed by NM, 3-Jun-2012.) (Revised by NM, 29-Jan-2013.)
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Theoremibi 163 Inference that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 17-Oct-2003.)
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Theoremibir 164 Inference that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 22-Jul-2004.)
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Theoremibd 165 Deduction that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 26-Jun-2004.)
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Theorempm5.74 166 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 167 Distribution of implication over biconditional (inference rule). (Contributed by NM, 1-Aug-1994.)
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Theorempm5.74ri 168 Distribution of implication over biconditional (reverse inference rule). (Contributed by NM, 1-Aug-1994.)
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Theorempm5.74d 169 Distribution of implication over biconditional (deduction rule). (Contributed by NM, 21-Mar-1996.)
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Theorempm5.74rd 170 Distribution of implication over biconditional (deduction rule). (Contributed by NM, 19-Mar-1997.)
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Theorembitri 171 An inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-Oct-2012.)
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Theorembitr2i 172 An inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.)
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Theorembitr3i 173 An inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.)
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Theorembitr4i 174 An inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.)
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Theorembitrd 175 Deduction form of bitri 171. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Apr-2013.)
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Theorembitr2d 176 Deduction form of bitr2i 172. (Contributed by NM, 9-Jun-2004.)
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Theorembitr3d 177 Deduction form of bitr3i 173. (Contributed by NM, 5-Aug-1993.)
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Theorembitr4d 178 Deduction form of bitr4i 174. (Contributed by NM, 5-Aug-1993.)
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Theoremsyl5bb 179 A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)
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Theoremsyl5rbb 180 A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)
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Theoremsyl5bbr 181 A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)
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Theoremsyl5rbbr 182 A syllogism inference from two biconditionals. (Contributed by NM, 25-Nov-1994.)
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Theoremsyl6bb 183 A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)
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Theoremsyl6rbb 184 A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)
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Theoremsyl6bbr 185 A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)
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Theoremsyl6rbbr 186 A syllogism inference from two biconditionals. (Contributed by NM, 25-Nov-1994.)
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Theorem3imtr3i 187 A mixed syllogism inference, useful for removing a definition from both sides of an implication. (Contributed by NM, 10-Aug-1994.)
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Theorem3imtr4i 188 A mixed syllogism inference, useful for applying a definition to both sides of an implication. (Contributed by NM, 5-Aug-1993.)
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Theorem3imtr3d 189 More general version of 3imtr3i 187. Useful for converting conditional definitions in a formula. (Contributed by NM, 8-Apr-1996.)
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Theorem3imtr4d 190 More general version of 3imtr4i 188. Useful for converting conditional definitions in a formula. (Contributed by NM, 26-Oct-1995.)
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Theorem3imtr3g 191 More general version of 3imtr3i 187. Useful for converting definitions in a formula. (Contributed by NM, 20-May-1996.) (Proof shortened by Wolf Lammen, 20-Dec-2013.)
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Theorem3imtr4g 192 More general version of 3imtr4i 188. Useful for converting definitions in a formula. (Contributed by NM, 20-May-1996.) (Proof shortened by Wolf Lammen, 20-Dec-2013.)
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Theorem3bitri 193 A chained inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.)
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Theorem3bitrri 194 A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.)
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Theorem3bitr2i 195 A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.)
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Theorem3bitr2ri 196 A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.)
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Theorem3bitr3i 197 A chained inference from transitive law for logical equivalence. (Contributed by NM, 19-Aug-1993.)
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Theorem3bitr3ri 198 A chained inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.)
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Theorem3bitr4i 199 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. (Contributed by NM, 5-Aug-1993.)
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Theorem3bitr4ri 200 A chained inference from transitive law for logical equivalence. (Contributed by NM, 2-Sep-1995.)
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