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Type | Label | Description |
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Statement | ||
Theorem | pm2.86 101 | Converse of Axiom ax-2 7. Theorem *2.86 of [WhiteheadRussell] p. 108. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 3-Apr-2013.) |
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Theorem | loolin 102 | The Linearity Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz. (Contributed by O'Cat, 12-Aug-2004.) |
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Theorem | loowoz 103 | 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.) |
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Syntax | wa 104 | Extend wff definition to include conjunction ('and'). |
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Syntax | wb 105 | Extend our wff definition to include the biconditional connective. |
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Axiom | ax-ia1 106 | Left 'and' elimination. One of the axioms of propositional logic. Use its alias simpl 109 instead for naming consistency with set.mm. (New usage is discouraged.) (Contributed by Mario Carneiro, 31-Jan-2015.) |
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Axiom | ax-ia2 107 | Right 'and' elimination. One of the axioms of propositional logic. (Contributed by Mario Carneiro, 31-Jan-2015.) Use its alias simpr 110 instead for naming consistency with set.mm. (New usage is discouraged.) |
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Axiom | ax-ia3 108 | 'And' introduction. One of the axioms of propositional logic. (Contributed by Mario Carneiro, 31-Jan-2015.) |
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Theorem | simpl 109 | 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.) |
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Theorem | simpr 110 | 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.) |
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Theorem | simpli 111 | Inference eliminating a conjunct. (Contributed by NM, 15-Jun-1994.) |
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Theorem | simpld 112 | Deduction eliminating a conjunct. (Contributed by NM, 5-Aug-1993.) |
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Theorem | simpri 113 | Inference eliminating a conjunct. (Contributed by NM, 15-Jun-1994.) |
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Theorem | simprd 114 | Deduction eliminating a conjunct. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Oct-2013.) |
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Theorem | ex 115 | 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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Theorem | expcom 116 | Exportation inference with commuted antecedents. (Contributed by NM, 25-May-2005.) |
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Definition | df-bi 117 |
This is our first definition, which introduces and defines the
biconditional connective ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() 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 979. 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 980) 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
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 117 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 388 which uses the biconditional instead. Other definitions of the biconditional, such as dfbi3dc 1397, only hold for decidable propositions, not all propositions. (Contributed by NM, 5-Aug-1993.) (Revised by Jim Kingdon, 24-Nov-2017.) |
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Theorem | biimp 118 | Property of the biconditional connective. (Contributed by NM, 11-May-1999.) (Revised by NM, 31-Jan-2015.) |
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Theorem | bi3 119 | Property of the biconditional connective. (Contributed by NM, 11-May-1999.) |
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Theorem | biimpi 120 | Infer an implication from a logical equivalence. (Contributed by NM, 5-Aug-1993.) |
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Theorem | sylbi 121 | A mixed syllogism inference from a biconditional and an implication. Useful for substituting an antecedent with a definition. (Contributed by NM, 3-Jan-1993.) |
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Theorem | sylib 122 | A mixed syllogism inference from an implication and a biconditional. (Contributed by NM, 3-Jan-1993.) |
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Theorem | sylbb 123 | A mixed syllogism inference from two biconditionals. (Contributed by BJ, 30-Mar-2019.) |
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Theorem | imp 124 | Importation inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.) |
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Theorem | impcom 125 | Importation inference with commuted antecedents. (Contributed by NM, 25-May-2005.) |
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Theorem | impbii 126 | Infer an equivalence from an implication and its converse. (Contributed by NM, 5-Aug-1993.) |
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Theorem | impbidd 127 | Deduce an equivalence from two implications. (Contributed by Rodolfo Medina, 12-Oct-2010.) |
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Theorem | impbid21d 128 | Deduce an equivalence from two implications. (Contributed by Wolf Lammen, 12-May-2013.) |
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Theorem | impbid 129 | Deduce an equivalence from two implications. (Contributed by NM, 5-Aug-1993.) (Revised by Wolf Lammen, 3-Nov-2012.) |
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Theorem | biimpr 130 | Property of the biconditional connective. (Contributed by NM, 11-May-1999.) (Proof shortened by Wolf Lammen, 11-Nov-2012.) |
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Theorem | bicom1 131 | Commutative law for equivalence. (Contributed by Wolf Lammen, 10-Nov-2012.) |
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Theorem | bicomi 132 | Inference from commutative law for logical equivalence. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 16-Sep-2013.) |
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Theorem | biimpri 133 | 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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Theorem | sylibr 134 | 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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Theorem | sylbir 135 | A mixed syllogism inference from a biconditional and an implication. (Contributed by NM, 5-Aug-1993.) |
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Theorem | sylbbr 136 |
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 121, sylib 122, sylbir 135, sylibr 134; four inferences inferring an implication from two biconditionals: sylbb 123, sylbbr 136, sylbb1 137, sylbb2 138; four inferences inferring a biconditional from two biconditionals: bitri 184, bitr2i 185, bitr3i 186, bitr4i 187 (and more for chaining more biconditionals). There are also closed forms and deduction versions of these, like, among many others, syld 45, syl5 32, syl6 33, mpbid 147, bitrd 188, bitrid 192, bitrdi 196 and variants. (Contributed by BJ, 21-Apr-2019.) |
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Theorem | sylbb1 137 | A mixed syllogism inference from two biconditionals. (Contributed by BJ, 21-Apr-2019.) |
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Theorem | sylbb2 138 | A mixed syllogism inference from two biconditionals. (Contributed by BJ, 21-Apr-2019.) |
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Theorem | pm3.2 139 | 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.) |
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Theorem | bicom 140 | Commutative law for equivalence. Theorem *4.21 of [WhiteheadRussell] p. 117. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 11-Nov-2012.) |
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Theorem | bicomd 141 | Commute two sides of a biconditional in a deduction. (Contributed by NM, 5-Aug-1993.) |
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Theorem | impbid1 142 | Infer an equivalence from two implications. (Contributed by NM, 6-Mar-2007.) |
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Theorem | impbid2 143 | Infer an equivalence from two implications. (Contributed by NM, 6-Mar-2007.) (Proof shortened by Wolf Lammen, 27-Sep-2013.) |
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Theorem | biimpd 144 | Deduce an implication from a logical equivalence. (Contributed by NM, 5-Aug-1993.) |
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Theorem | mpbi 145 | An inference from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.) |
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Theorem | mpbir 146 | An inference from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.) |
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Theorem | mpbid 147 | A deduction from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.) |
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Theorem | mpbii 148 | 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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Theorem | sylibd 149 | A syllogism deduction. (Contributed by NM, 3-Aug-1994.) |
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Theorem | sylbid 150 | A syllogism deduction. (Contributed by NM, 3-Aug-1994.) |
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Theorem | mpbidi 151 | A deduction from a biconditional, related to modus ponens. (Contributed by NM, 9-Aug-1994.) |
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Theorem | biimtrid 152 | 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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Theorem | biimtrrid 153 | A mixed syllogism inference from a nested implication and a biconditional. (Contributed by NM, 5-Aug-1993.) |
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Theorem | imbitrid 154 | A mixed syllogism inference. (Contributed by NM, 12-Jan-1993.) |
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Theorem | syl5ibcom 155 | A mixed syllogism inference. (Contributed by NM, 19-Jun-2007.) |
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Theorem | imbitrrid 156 | A mixed syllogism inference. (Contributed by NM, 3-Apr-1994.) (Revised by NM, 22-Sep-2013.) |
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Theorem | syl5ibrcom 157 | A mixed syllogism inference. (Contributed by NM, 20-Jun-2007.) |
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Theorem | biimprd 158 | 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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Theorem | biimpcd 159 | 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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Theorem | biimprcd 160 | 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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Theorem | imbitrdi 161 | A mixed syllogism inference from a nested implication and a biconditional. (Contributed by NM, 5-Aug-1993.) |
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Theorem | imbitrrdi 162 | 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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Theorem | biimtrdi 163 | A mixed syllogism inference. (Contributed by NM, 2-Jan-1994.) |
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Theorem | syl6bir 164 | A mixed syllogism inference. (Contributed by NM, 18-May-1994.) |
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Theorem | syl7bi 165 | A mixed syllogism inference from a doubly nested implication and a biconditional. (Contributed by NM, 5-Aug-1993.) |
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Theorem | syl8ib 166 | 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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Theorem | mpbird 167 | A deduction from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.) |
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Theorem | mpbiri 168 | 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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Theorem | sylibrd 169 | A syllogism deduction. (Contributed by NM, 3-Aug-1994.) |
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Theorem | sylbird 170 | A syllogism deduction. (Contributed by NM, 3-Aug-1994.) |
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Theorem | biid 171 | Principle of identity for logical equivalence. Theorem *4.2 of [WhiteheadRussell] p. 117. (Contributed by NM, 5-Aug-1993.) |
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Theorem | biidd 172 | Principle of identity with antecedent. (Contributed by NM, 25-Nov-1995.) |
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Theorem | pm5.1im 173 |
Two propositions are equivalent if they are both true. Closed form of
2th 174. Equivalent to a biimp 118-like version of the xor-connective.
This theorem stays true, no matter how you permute its operands. This is
evident from its sharper version ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | 2th 174 | Two truths are equivalent. (Contributed by NM, 18-Aug-1993.) |
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Theorem | 2thd 175 | Two truths are equivalent (deduction form). (Contributed by NM, 3-Jun-2012.) (Revised by NM, 29-Jan-2013.) |
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Theorem | ibi 176 | Inference that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 17-Oct-2003.) |
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Theorem | ibir 177 | Inference that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 22-Jul-2004.) |
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Theorem | ibd 178 | Deduction that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 26-Jun-2004.) |
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Theorem | pm5.74 179 | 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.) |
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Theorem | pm5.74i 180 | Distribution of implication over biconditional (inference form). (Contributed by NM, 1-Aug-1994.) |
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Theorem | pm5.74ri 181 | Distribution of implication over biconditional (reverse inference form). (Contributed by NM, 1-Aug-1994.) |
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Theorem | pm5.74d 182 | Distribution of implication over biconditional (deduction form). (Contributed by NM, 21-Mar-1996.) |
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Theorem | pm5.74rd 183 | Distribution of implication over biconditional (deduction form). (Contributed by NM, 19-Mar-1997.) |
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Theorem | bitri 184 | 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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Theorem | bitr2i 185 | An inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.) |
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Theorem | bitr3i 186 | An inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.) |
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Theorem | bitr4i 187 | An inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.) |
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Theorem | bitrd 188 | Deduction form of bitri 184. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Apr-2013.) |
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Theorem | bitr2d 189 | Deduction form of bitr2i 185. (Contributed by NM, 9-Jun-2004.) |
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Theorem | bitr3d 190 | Deduction form of bitr3i 186. (Contributed by NM, 5-Aug-1993.) |
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Theorem | bitr4d 191 | Deduction form of bitr4i 187. (Contributed by NM, 5-Aug-1993.) |
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Theorem | bitrid 192 | A syllogism inference from two biconditionals. (Contributed by NM, 12-Mar-1993.) |
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Theorem | bitr2id 193 | A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.) |
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Theorem | bitr3id 194 | A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.) |
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Theorem | bitr3di 195 | A syllogism inference from two biconditionals. (Contributed by NM, 25-Nov-1994.) |
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Theorem | bitrdi 196 | A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.) |
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Theorem | bitr2di 197 | A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.) |
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Theorem | bitr4di 198 | A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.) |
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Theorem | bitr4id 199 | A syllogism inference from two biconditionals. (Contributed by NM, 25-Nov-1994.) |
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Theorem | 3imtr3i 200 | 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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