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Type | Label | Description |
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Statement | ||
Theorem | pm2.24 101 | Theorem *2.24 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm2.18 102 | Proof by contradiction. Theorem *2.18 of [WhiteheadRussell] p. 103. Also called the Law of Clavius. (Contributed by NM, 5-Aug-1993.) |
Theorem | pm2.18d 103 | Deduction based on reductio ad absurdum. (Contributed by FL, 12-Jul-2009.) (Proof shortened by Andrew Salmon, 7-May-2011.) |
Theorem | notnot2 104 | Converse of double negation. Theorem *2.14 of [WhiteheadRussell] p. 102. (Contributed by NM, 5-Aug-1993.) (Proof shortened by David Harvey, 5-Sep-1999.) (Proof shortened by Josh Purinton, 29-Dec-2000.) |
Theorem | notnotrd 105 | Deduction converting double-negation into the original wff, aka the double negation rule. A translation of natural deduction rule -C, Gamma => Gamma ; see natded in set.mm. This is definition NNC in [Pfenning] p. 17. This rule is valid in classical logic (which MPE uses), but not intuitionistic logic. (Contributed by DAW, 8-Feb-2017.) |
Theorem | notnotri 106 | Inference from double negation. (Contributed by NM, 27-Feb-2008.) |
Theorem | con2d 107 | A contraposition deduction. (Contributed by NM, 19-Aug-1993.) |
Theorem | con2 108 | Contraposition. Theorem *2.03 of [WhiteheadRussell] p. 100. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Feb-2013.) |
Theorem | mt2d 109 | Modus tollens deduction. (Contributed by NM, 4-Jul-1994.) |
Theorem | mt2i 110 | Modus tollens inference. (Contributed by NM, 26-Mar-1995.) (Proof shortened by Wolf Lammen, 15-Sep-2012.) |
Theorem | nsyl3 111 | A negated syllogism inference. (Contributed by NM, 1-Dec-1995.) |
Theorem | con2i 112 | A contraposition inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 28-Nov-2008.) (Proof shortened by Wolf Lammen, 13-Jun-2013.) |
Theorem | nsyl 113 | A negated syllogism inference. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Wolf Lammen, 2-Mar-2013.) |
Theorem | notnot1 114 | Converse of double negation. Theorem *2.12 of [WhiteheadRussell] p. 101. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 2-Mar-2013.) |
Theorem | notnoti 115 | Infer double negation. (Contributed by NM, 27-Feb-2008.) |
Theorem | con1d 116 | A contraposition deduction. (Contributed by NM, 5-Aug-1993.) |
Theorem | mt3d 117 | Modus tollens deduction. (Contributed by NM, 26-Mar-1995.) |
Theorem | mt3i 118 | Modus tollens inference. (Contributed by NM, 26-Mar-1995.) (Proof shortened by Wolf Lammen, 15-Sep-2012.) |
Theorem | nsyl2 119 | A negated syllogism inference. (Contributed by NM, 26-Jun-1994.) |
Theorem | con1 120 | Contraposition. Theorem *2.15 of [WhiteheadRussell] p. 102. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Feb-2013.) |
Theorem | con1i 121 | A contraposition inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 28-Nov-2008.) (Proof shortened by Wolf Lammen, 19-Jun-2013.) |
Theorem | con4i 122 | Inference rule derived from Axiom ax-3 8. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 21-Jun-2013.) |
Theorem | pm2.21i 123 | A contradiction implies anything. Inference from pm2.21 100. (Contributed by NM, 16-Sep-1993.) |
Theorem | pm2.24ii 124 | A contradiction implies anything. Inference from pm2.24 101. (Contributed by NM, 27-Feb-2008.) |
Theorem | con3d 125 | A contraposition deduction. (Contributed by NM, 5-Aug-1993.) |
Theorem | con3 126 | Contraposition. Theorem *2.16 of [WhiteheadRussell] p. 103. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-Feb-2013.) |
Theorem | con3i 127 | A contraposition inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 20-Jun-2013.) |
Theorem | con3rr3 128 | Rotate through consequent right. (Contributed by Wolf Lammen, 3-Nov-2013.) |
Theorem | mt4 129 | The rule of modus tollens. (Contributed by Wolf Lammen, 12-May-2013.) |
Theorem | mt4d 130 | Modus tollens deduction. (Contributed by NM, 9-Jun-2006.) |
Theorem | mt4i 131 | Modus tollens inference. (Contributed by Wolf Lammen, 12-May-2013.) |
Theorem | nsyld 132 | A negated syllogism deduction. (Contributed by NM, 9-Apr-2005.) |
Theorem | nsyli 133 | A negated syllogism inference. (Contributed by NM, 3-May-1994.) |
Theorem | nsyl4 134 | A negated syllogism inference. (Contributed by NM, 15-Feb-1996.) |
Theorem | pm2.24d 135 | Deduction version of pm2.24 101. (Contributed by NM, 30-Jan-2006.) |
Theorem | pm2.24i 136 | Inference version of pm2.24 101. (Contributed by NM, 20-Aug-2001.) |
Theorem | pm3.2im 137 | Theorem *3.2 of [WhiteheadRussell] p. 111, expressed with primitive connectives. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.) |
Theorem | mth8 138 | Theorem 8 of [Margaris] p. 60. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.) |
Theorem | jc 139 | Inference joining the consequents of two premises. (Contributed by NM, 5-Aug-1993.) |
Theorem | impi 140 | An importation inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 20-Jul-2013.) |
Theorem | expi 141 | An exportation inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 28-Nov-2008.) |
Theorem | simprim 142 | Simplification. Similar to Theorem *3.27 (Simp) of [WhiteheadRussell] p. 112. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-Nov-2012.) |
Theorem | simplim 143 | Simplification. Similar to Theorem *3.26 (Simp) of [WhiteheadRussell] p. 112. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 21-Jul-2012.) |
Theorem | pm2.5 144 | Theorem *2.5 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 9-Oct-2012.) |
Theorem | pm2.51 145 | Theorem *2.51 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm2.521 146 | Theorem *2.521 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 8-Oct-2012.) |
Theorem | pm2.52 147 | Theorem *2.52 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 8-Oct-2012.) |
Theorem | expt 148 | Exportation theorem expressed with primitive connectives. (Contributed by NM, 5-Aug-1993.) |
Theorem | impt 149 | Importation theorem expressed with primitive connectives. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 20-Jul-2013.) |
Theorem | pm2.61d 150 | Deduction eliminating an antecedent. (Contributed by NM, 27-Apr-1994.) (Proof shortened by Wolf Lammen, 12-Sep-2013.) |
Theorem | pm2.61d1 151 | Inference eliminating an antecedent. (Contributed by NM, 15-Jul-2005.) |
Theorem | pm2.61d2 152 | Inference eliminating an antecedent. (Contributed by NM, 18-Aug-1993.) |
Theorem | ja 153 | Inference joining the antecedents of two premises. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 19-Feb-2008.) |
Theorem | jad 154 | Deduction form of ja 153. (Contributed by Scott Fenton, 13-Dec-2010.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
Theorem | jarl 155 | Elimination of a nested antecedent as a kind of reversal of inference ja 153. (Contributed by Wolf Lammen, 10-May-2013.) |
Theorem | pm2.61i 156 | Inference eliminating an antecedent. (Contributed by NM, 5-Apr-1994.) (Proof shortened by Wolf Lammen, 12-Sep-2013.) |
Theorem | pm2.61ii 157 | Inference eliminating two antecedents. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.) |
Theorem | pm2.61nii 158 | Inference eliminating two antecedents. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 13-Nov-2012.) |
Theorem | pm2.61iii 159 | Inference eliminating three antecedents. (Contributed by NM, 2-Jan-2002.) (Proof shortened by Wolf Lammen, 22-Sep-2013.) |
Theorem | pm2.01 160 | Reductio ad absurdum. Theorem *2.01 of [WhiteheadRussell] p. 100. (Contributed by NM, 18-Aug-1993.) (Proof shortened by O'Cat, 21-Nov-2008.) (Proof shortened by Wolf Lammen, 31-Oct-2012.) |
Theorem | pm2.01d 161 | Deduction based on reductio ad absurdum. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 5-Mar-2013.) |
Theorem | pm2.6 162 | Theorem *2.6 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm2.61 163 | Theorem *2.61 of [WhiteheadRussell] p. 107. Useful for eliminating an antecedent. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 22-Sep-2013.) |
Theorem | pm2.65 164 | Theorem *2.65 of [WhiteheadRussell] p. 107. Proof by contradiction. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 8-Mar-2013.) |
Theorem | pm2.65i 165 | Inference rule for proof by contradiction. (Contributed by NM, 18-May-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.) |
Theorem | pm2.65d 166 | Deduction rule for proof by contradiction. (Contributed by NM, 26-Jun-1994.) (Proof shortened by Wolf Lammen, 26-May-2013.) |
Theorem | mto 167 | The rule of modus tollens. The rule says, "if is not true, and implies , then must also be not true." Modus tollens is short for "modus tollendo tollens," a Latin phrase that means "the mood that by denying affirms" [Sanford] p. 39. It is also called denying the consequent. Modus tollens is closely related to modus ponens ax-mp 5. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 11-Sep-2013.) |
Theorem | mtod 168 | Modus tollens deduction. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.) |
Theorem | mtoi 169 | Modus tollens inference. (Contributed by NM, 5-Jul-1994.) (Proof shortened by Wolf Lammen, 15-Sep-2012.) |
Theorem | mt2 170 | A rule similar to modus tollens. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 10-Sep-2013.) |
Theorem | mt3 171 | A rule similar to modus tollens. (Contributed by NM, 18-May-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.) |
Theorem | peirce 172 | Peirce's axiom. This odd-looking theorem is the "difference" between an intuitionistic system of propositional calculus and a classical system and is not accepted by intuitionists. When Peirce's axiom is added to an intuitionistic system, the system becomes equivalent to our classical system ax-1 6 through ax-3 8. A curious fact about this theorem is that it requires ax-3 8 for its proof even though the result has no negation connectives in it. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 9-Oct-2012.) |
Theorem | loolin 173 | The Linearity Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz. For a version not using ax-3 8, see loolinALT 95. (Contributed by O'Cat, 12-Aug-2004.) (Proof shortened by Wolf Lammen, 2-Nov-2012.) |
Theorem | looinv 174 | The Inversion Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz. Using dfor2 400, we can see that this essentially expresses "disjunction commutes." Theorem *2.69 of [WhiteheadRussell] p. 108. (Contributed by NM, 12-Aug-2004.) |
Theorem | bijust 175 | Theorem used to justify definition of biconditional df-bi 177. (Contributed by NM, 11-May-1999.) (Proof shortened by Josh Purinton, 29-Dec-2000.) |
Definition df-bi 177 in this section 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 359 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. | ||
Syntax | wb 176 | Extend our wff definition to include the biconditional connective. |
Definition | df-bi 177 |
Define the biconditional (logical 'iff').
Definition df-bi 177 in this section 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 359 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 936) 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 175. It is impossible to use df-bi 177 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 177 in the proof with the corresponding bijust 175 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 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 a naming convention for human readability.) After we define the constant true (df-tru 1319) and the constant false (df-fal 1320), we will be able to prove these truth table values: (trubitru 1350), (trubifal 1351), (falbitru 1352), and (falbifal 1353). See dfbi1 184, dfbi2 609, and dfbi3 863 for theorems suggesting typical textbook definitions of , showing that our definition has the properties we expect. Theorem dfbi1 184 is particularly useful if we want to eliminate from an expression to convert it to primitives. Theorem dfbi 610 shows this definition rewritten in an abbreviated form after conjunction is introduced, for easier understanding. Contrast with (df-or 359), (wi 4), (df-nan 1288), and (df-xor 1305) . In some sense returns true if two truth values are equal; (df-cleq 2346) returns true if two classes are equal. (Contributed by NM, 5-Aug-1993.) |
Theorem | bi1 178 | Property of the biconditional connective. (Contributed by NM, 11-May-1999.) |
Theorem | bi3 179 | Property of the biconditional connective. (Contributed by NM, 11-May-1999.) |
Theorem | impbii 180 | Infer an equivalence from an implication and its converse. (Contributed by NM, 5-Aug-1993.) |
Theorem | impbidd 181 | Deduce an equivalence from two implications. (Contributed by Rodolfo Medina, 12-Oct-2010.) |
Theorem | impbid21d 182 | Deduce an equivalence from two implications. (Contributed by Wolf Lammen, 12-May-2013.) |
Theorem | impbid 183 | Deduce an equivalence from two implications. (Contributed by NM, 5-Aug-1993.) (Revised by Wolf Lammen, 3-Nov-2012.) |
Theorem | dfbi1 184 | Relate the biconditional connective to primitive connectives. See dfbi1gb 185 for an unusual version proved directly from axioms. (Contributed by NM, 5-Aug-1993.) |
Theorem | dfbi1gb 185 | This proof of dfbi1 184, discovered by Gregory Bush on 8-Mar-2004, has several curious properties. First, it has only 17 steps directly from the axioms and df-bi 177, compared to over 800 steps were the proof of dfbi1 184 expanded into axioms. Second, step 2 demands only the property of "true"; any axiom (or theorem) could be used. It might be thought, therefore, that it is in some sense redundant, but in fact no proof is shorter than this (measured by number of steps). Third, it illustrates how intermediate steps can "blow up" in size even in short proofs. Fourth, the compressed proof is only 182 bytes (or 17 bytes in D-proof notation), but the generated web page is over 200kB with intermediate steps that are essentially incomprehensible to humans (other than Gregory Bush). If there were an obfuscated code contest for proofs, this would be a contender. This "blowing up" and incomprehensibility of the intermediate steps vividly demonstrate the advantages of using many layered intermediate theorems, since each theorem is easier to understand. (Contributed by Gregory Bush, 10-Mar-2004.) (New usage is discouraged.) (Proof modification is discouraged.) |
Theorem | biimpi 186 | Infer an implication from a logical equivalence. (Contributed by NM, 5-Aug-1993.) |
Theorem | sylbi 187 | 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 | sylib 188 | A mixed syllogism inference from an implication and a biconditional. (Contributed by NM, 5-Aug-1993.) |
Theorem | bi2 189 | Property of the biconditional connective. (Contributed by NM, 11-May-1999.) (Proof shortened by Wolf Lammen, 11-Nov-2012.) |
Theorem | bicom1 190 | Commutative law for equivalence. (Contributed by Wolf Lammen, 10-Nov-2012.) |
Theorem | bicom 191 | Commutative law for equivalence. Theorem *4.21 of [WhiteheadRussell] p. 117. (Contributed by NM, 5-Aug-1993.) |
Theorem | bicomd 192 | Commute two sides of a biconditional in a deduction. (Contributed by NM, 5-Aug-1993.) |
Theorem | bicomi 193 | Inference from commutative law for logical equivalence. (Contributed by NM, 5-Aug-1993.) |
Theorem | impbid1 194 | Infer an equivalence from two implications. (Contributed by NM, 6-Mar-2007.) |
Theorem | impbid2 195 | Infer an equivalence from two implications. (Contributed by NM, 6-Mar-2007.) (Proof shortened by Wolf Lammen, 27-Sep-2013.) |
Theorem | impcon4bid 196 | A variation on impbid 183 with contraposition. (Contributed by Jeff Hankins, 3-Jul-2009.) |
Theorem | biimpri 197 | Infer a converse implication from a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 16-Sep-2013.) |
Theorem | biimpd 198 | Deduce an implication from a logical equivalence. (Contributed by NM, 5-Aug-1993.) |
Theorem | mpbi 199 | An inference from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.) |
Theorem | mpbir 200 | An inference from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.) |
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