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Type | Label | Description |
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Statement | ||

Theorem | simpr 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.) |

Theorem | simpli 102 | Inference eliminating a conjunct. (Contributed by NM, 15-Jun-1994.) |

Theorem | simpld 103 | Deduction eliminating a conjunct. (Contributed by NM, 5-Aug-1993.) |

Theorem | simpri 104 | Inference eliminating a conjunct. (Contributed by NM, 15-Jun-1994.) |

Theorem | simprd 105 | Deduction eliminating a conjunct. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Oct-2013.) |

Theorem | ex 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.) |

Theorem | expcom 107 | Exportation inference with commuted antecedents. (Contributed by NM, 25-May-2005.) |

Definition | df-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.) |

Theorem | bi1 109 | Property of the biconditional connective. (Contributed by NM, 11-May-1999.) (Revised by NM, 31-Jan-2015.) |

Theorem | bi3 110 | Property of the biconditional connective. (Contributed by NM, 11-May-1999.) |

Theorem | biimpi 111 | Infer an implication from a logical equivalence. (Contributed by NM, 5-Aug-1993.) |

Theorem | sylbi 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.) |

Theorem | imp 113 | Importation inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.) |

Theorem | impcom 114 | Importation inference with commuted antecedents. (Contributed by NM, 25-May-2005.) |

Theorem | impbii 115 | Infer an equivalence from an implication and its converse. (Contributed by NM, 5-Aug-1993.) |

Theorem | impbidd 116 | Deduce an equivalence from two implications. (Contributed by Rodolfo Medina, 12-Oct-2010.) |

Theorem | impbid21d 117 | Deduce an equivalence from two implications. (Contributed by Wolf Lammen, 12-May-2013.) |

Theorem | impbid 118 | Deduce an equivalence from two implications. (Contributed by NM, 5-Aug-1993.) (Revised by Wolf Lammen, 3-Nov-2012.) |

Theorem | bi2 119 | Property of the biconditional connective. (Contributed by NM, 11-May-1999.) (Proof shortened by Wolf Lammen, 11-Nov-2012.) |

Theorem | bicom1 120 | Commutative law for equivalence. (Contributed by Wolf Lammen, 10-Nov-2012.) |

Theorem | bicomi 121 | Inference from commutative law for logical equivalence. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 16-Sep-2013.) |

Theorem | biimpri 122 | Infer a converse implication from a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 16-Sep-2013.) |

Theorem | sylbir 123 | A mixed syllogism inference from a biconditional and an implication. (Contributed by NM, 5-Aug-1993.) |

Theorem | pm3.2 124 | 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.) |

Theorem | sylib 125 | A mixed syllogism inference from an implication and a biconditional. (Contributed by NM, 5-Aug-1993.) |

Theorem | bicom 126 | 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 127 | Commute two sides of a biconditional in a deduction. (Contributed by NM, 5-Aug-1993.) |

Theorem | impbid1 128 | Infer an equivalence from two implications. (Contributed by NM, 6-Mar-2007.) |

Theorem | impbid2 129 | Infer an equivalence from two implications. (Contributed by NM, 6-Mar-2007.) (Proof shortened by Wolf Lammen, 27-Sep-2013.) |

Theorem | biimpd 130 | Deduce an implication from a logical equivalence. (Contributed by NM, 5-Aug-1993.) |

Theorem | mpbi 131 | An inference from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.) |

Theorem | mpbir 132 | An inference from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.) |

Theorem | mpbid 133 | A deduction from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.) |

Theorem | mpbii 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.) |

Theorem | sylibr 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.) |

Theorem | sylibd 136 | A syllogism deduction. (Contributed by NM, 3-Aug-1994.) |

Theorem | sylbid 137 | A syllogism deduction. (Contributed by NM, 3-Aug-1994.) |

Theorem | mpbidi 138 | A deduction from a biconditional, related to modus ponens. (Contributed by NM, 9-Aug-1994.) |

Theorem | syl5bi 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.) |

Theorem | syl5bir 140 | A mixed syllogism inference from a nested implication and a biconditional. (Contributed by NM, 5-Aug-1993.) |

Theorem | syl5ib 141 | A mixed syllogism inference. (Contributed by NM, 5-Aug-1993.) |

Theorem | syl5ibcom 142 | A mixed syllogism inference. (Contributed by NM, 19-Jun-2007.) |

Theorem | syl5ibr 143 | A mixed syllogism inference. (Contributed by NM, 3-Apr-1994.) (Revised by NM, 22-Sep-2013.) |

Theorem | syl5ibrcom 144 | A mixed syllogism inference. (Contributed by NM, 20-Jun-2007.) |

Theorem | biimprd 145 | Deduce a converse implication from a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 22-Sep-2013.) |

Theorem | biimpcd 146 | Deduce a commuted implication from a logical equivalence. (Contributed by NM, 3-May-1994.) (Proof shortened by Wolf Lammen, 22-Sep-2013.) |

Theorem | biimprcd 147 | 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 148 | A mixed syllogism inference from a nested implication and a biconditional. (Contributed by NM, 5-Aug-1993.) |

Theorem | syl6ibr 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.) |

Theorem | syl6bi 150 | A mixed syllogism inference. (Contributed by NM, 2-Jan-1994.) |

Theorem | syl6bir 151 | A mixed syllogism inference. (Contributed by NM, 18-May-1994.) |

Theorem | syl7bi 152 | A mixed syllogism inference from a doubly nested implication and a biconditional. (Contributed by NM, 5-Aug-1993.) |

Theorem | syl8ib 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.) |

Theorem | mpbird 154 | A deduction from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.) |

Theorem | mpbiri 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.) |

Theorem | sylibrd 156 | A syllogism deduction. (Contributed by NM, 3-Aug-1994.) |

Theorem | sylbird 157 | A syllogism deduction. (Contributed by NM, 3-Aug-1994.) |

Theorem | biid 158 | Principle of identity for logical equivalence. Theorem *4.2 of [WhiteheadRussell] p. 117. (Contributed by NM, 5-Aug-1993.) |

Theorem | biidd 159 | Principle of identity with antecedent. (Contributed by NM, 25-Nov-1995.) |

Theorem | pm5.1im 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.) |

Theorem | 2th 161 | Two truths are equivalent. (Contributed by NM, 18-Aug-1993.) |

Theorem | 2thd 162 | Two truths are equivalent (deduction rule). (Contributed by NM, 3-Jun-2012.) (Revised by NM, 29-Jan-2013.) |

Theorem | ibi 163 | Inference that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 17-Oct-2003.) |

Theorem | ibir 164 | Inference that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 22-Jul-2004.) |

Theorem | ibd 165 | Deduction that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 26-Jun-2004.) |

Theorem | pm5.74 166 | 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.) |

Theorem | pm5.74i 167 | Distribution of implication over biconditional (inference rule). (Contributed by NM, 1-Aug-1994.) |

Theorem | pm5.74ri 168 | Distribution of implication over biconditional (reverse inference rule). (Contributed by NM, 1-Aug-1994.) |

Theorem | pm5.74d 169 | Distribution of implication over biconditional (deduction rule). (Contributed by NM, 21-Mar-1996.) |

Theorem | pm5.74rd 170 | Distribution of implication over biconditional (deduction rule). (Contributed by NM, 19-Mar-1997.) |

Theorem | bitri 171 | An inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-Oct-2012.) |

Theorem | bitr2i 172 | An inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.) |

Theorem | bitr3i 173 | An inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.) |

Theorem | bitr4i 174 | An inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.) |

Theorem | bitrd 175 | Deduction form of bitri 171. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Apr-2013.) |

Theorem | bitr2d 176 | Deduction form of bitr2i 172. (Contributed by NM, 9-Jun-2004.) |

Theorem | bitr3d 177 | Deduction form of bitr3i 173. (Contributed by NM, 5-Aug-1993.) |

Theorem | bitr4d 178 | Deduction form of bitr4i 174. (Contributed by NM, 5-Aug-1993.) |

Theorem | syl5bb 179 | A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.) |

Theorem | syl5rbb 180 | A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.) |

Theorem | syl5bbr 181 | A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.) |

Theorem | syl5rbbr 182 | A syllogism inference from two biconditionals. (Contributed by NM, 25-Nov-1994.) |

Theorem | syl6bb 183 | A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.) |

Theorem | syl6rbb 184 | A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.) |

Theorem | syl6bbr 185 | A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.) |

Theorem | syl6rbbr 186 | A syllogism inference from two biconditionals. (Contributed by NM, 25-Nov-1994.) |

Theorem | 3imtr3i 187 | A mixed syllogism inference, useful for removing a definition from both sides of an implication. (Contributed by NM, 10-Aug-1994.) |

Theorem | 3imtr4i 188 | A mixed syllogism inference, useful for applying a definition to both sides of an implication. (Contributed by NM, 5-Aug-1993.) |

Theorem | 3imtr3d 189 | More general version of 3imtr3i 187. Useful for converting conditional definitions in a formula. (Contributed by NM, 8-Apr-1996.) |

Theorem | 3imtr4d 190 | More general version of 3imtr4i 188. Useful for converting conditional definitions in a formula. (Contributed by NM, 26-Oct-1995.) |

Theorem | 3imtr3g 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.) |

Theorem | 3imtr4g 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.) |

Theorem | 3bitri 193 | A chained inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.) |

Theorem | 3bitrri 194 | A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.) |

Theorem | 3bitr2i 195 | A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.) |

Theorem | 3bitr2ri 196 | A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.) |

Theorem | 3bitr3i 197 | A chained inference from transitive law for logical equivalence. (Contributed by NM, 19-Aug-1993.) |

Theorem | 3bitr3ri 198 | A chained inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.) |

Theorem | 3bitr4i 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.) |

Theorem | 3bitr4ri 200 | A chained inference from transitive law for logical equivalence. (Contributed by NM, 2-Sep-1995.) |

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