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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | mpjao3dan 1201 | Eliminate a 3-way disjunction in a deduction. (Contributed by Thierry Arnoux, 13-Apr-2018.) |
⊢ ((φ ∧ ψ) → χ) & ⊢ ((φ ∧ θ) → χ) & ⊢ ((φ ∧ τ) → χ) & ⊢ (φ → (ψ ∨ θ ∨ τ)) ⇒ ⊢ (φ → χ) | ||
Theorem | 3jaao 1202 | Inference conjoining and disjoining the antecedents of three implications. (Contributed by Jeff Hankins, 15-Aug-2009.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
⊢ (φ → (ψ → χ)) & ⊢ (θ → (τ → χ)) & ⊢ (η → (ζ → χ)) ⇒ ⊢ ((φ ∧ θ ∧ η) → ((ψ ∨ τ ∨ ζ) → χ)) | ||
Theorem | 3ianorr 1203 | Triple disjunction implies negated triple conjunction. (Contributed by Jim Kingdon, 23-Dec-2018.) |
⊢ ((¬ φ ∨ ¬ ψ ∨ ¬ χ) → ¬ (φ ∧ ψ ∧ χ)) | ||
Theorem | syl3an9b 1204 | Nested syllogism inference conjoining 3 dissimilar antecedents. (Contributed by NM, 1-May-1995.) |
⊢ (φ → (ψ ↔ χ)) & ⊢ (θ → (χ ↔ τ)) & ⊢ (η → (τ ↔ ζ)) ⇒ ⊢ ((φ ∧ θ ∧ η) → (ψ ↔ ζ)) | ||
Theorem | 3orbi123d 1205 | Deduction joining 3 equivalences to form equivalence of disjunctions. (Contributed by NM, 20-Apr-1994.) |
⊢ (φ → (ψ ↔ χ)) & ⊢ (φ → (θ ↔ τ)) & ⊢ (φ → (η ↔ ζ)) ⇒ ⊢ (φ → ((ψ ∨ θ ∨ η) ↔ (χ ∨ τ ∨ ζ))) | ||
Theorem | 3anbi123d 1206 | Deduction joining 3 equivalences to form equivalence of conjunctions. (Contributed by NM, 22-Apr-1994.) |
⊢ (φ → (ψ ↔ χ)) & ⊢ (φ → (θ ↔ τ)) & ⊢ (φ → (η ↔ ζ)) ⇒ ⊢ (φ → ((ψ ∧ θ ∧ η) ↔ (χ ∧ τ ∧ ζ))) | ||
Theorem | 3anbi12d 1207 | Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.) |
⊢ (φ → (ψ ↔ χ)) & ⊢ (φ → (θ ↔ τ)) ⇒ ⊢ (φ → ((ψ ∧ θ ∧ η) ↔ (χ ∧ τ ∧ η))) | ||
Theorem | 3anbi13d 1208 | Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.) |
⊢ (φ → (ψ ↔ χ)) & ⊢ (φ → (θ ↔ τ)) ⇒ ⊢ (φ → ((ψ ∧ η ∧ θ) ↔ (χ ∧ η ∧ τ))) | ||
Theorem | 3anbi23d 1209 | Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.) |
⊢ (φ → (ψ ↔ χ)) & ⊢ (φ → (θ ↔ τ)) ⇒ ⊢ (φ → ((η ∧ ψ ∧ θ) ↔ (η ∧ χ ∧ τ))) | ||
Theorem | 3anbi1d 1210 | Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.) |
⊢ (φ → (ψ ↔ χ)) ⇒ ⊢ (φ → ((ψ ∧ θ ∧ τ) ↔ (χ ∧ θ ∧ τ))) | ||
Theorem | 3anbi2d 1211 | Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.) |
⊢ (φ → (ψ ↔ χ)) ⇒ ⊢ (φ → ((θ ∧ ψ ∧ τ) ↔ (θ ∧ χ ∧ τ))) | ||
Theorem | 3anbi3d 1212 | Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.) |
⊢ (φ → (ψ ↔ χ)) ⇒ ⊢ (φ → ((θ ∧ τ ∧ ψ) ↔ (θ ∧ τ ∧ χ))) | ||
Theorem | 3anim123d 1213 | Deduction joining 3 implications to form implication of conjunctions. (Contributed by NM, 24-Feb-2005.) |
⊢ (φ → (ψ → χ)) & ⊢ (φ → (θ → τ)) & ⊢ (φ → (η → ζ)) ⇒ ⊢ (φ → ((ψ ∧ θ ∧ η) → (χ ∧ τ ∧ ζ))) | ||
Theorem | 3orim123d 1214 | Deduction joining 3 implications to form implication of disjunctions. (Contributed by NM, 4-Apr-1997.) |
⊢ (φ → (ψ → χ)) & ⊢ (φ → (θ → τ)) & ⊢ (φ → (η → ζ)) ⇒ ⊢ (φ → ((ψ ∨ θ ∨ η) → (χ ∨ τ ∨ ζ))) | ||
Theorem | an6 1215 | Rearrangement of 6 conjuncts. (Contributed by NM, 13-Mar-1995.) |
⊢ (((φ ∧ ψ ∧ χ) ∧ (θ ∧ τ ∧ η)) ↔ ((φ ∧ θ) ∧ (ψ ∧ τ) ∧ (χ ∧ η))) | ||
Theorem | 3an6 1216 | Analog of an4 520 for triple conjunction. (Contributed by Scott Fenton, 16-Mar-2011.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (((φ ∧ ψ) ∧ (χ ∧ θ) ∧ (τ ∧ η)) ↔ ((φ ∧ χ ∧ τ) ∧ (ψ ∧ θ ∧ η))) | ||
Theorem | 3or6 1217 | Analog of or4 687 for triple conjunction. (Contributed by Scott Fenton, 16-Mar-2011.) |
⊢ (((φ ∨ ψ) ∨ (χ ∨ θ) ∨ (τ ∨ η)) ↔ ((φ ∨ χ ∨ τ) ∨ (ψ ∨ θ ∨ η))) | ||
Theorem | mp3an1 1218 | An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.) |
⊢ φ & ⊢ ((φ ∧ ψ ∧ χ) → θ) ⇒ ⊢ ((ψ ∧ χ) → θ) | ||
Theorem | mp3an2 1219 | An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.) |
⊢ ψ & ⊢ ((φ ∧ ψ ∧ χ) → θ) ⇒ ⊢ ((φ ∧ χ) → θ) | ||
Theorem | mp3an3 1220 | An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.) |
⊢ χ & ⊢ ((φ ∧ ψ ∧ χ) → θ) ⇒ ⊢ ((φ ∧ ψ) → θ) | ||
Theorem | mp3an12 1221 | An inference based on modus ponens. (Contributed by NM, 13-Jul-2005.) |
⊢ φ & ⊢ ψ & ⊢ ((φ ∧ ψ ∧ χ) → θ) ⇒ ⊢ (χ → θ) | ||
Theorem | mp3an13 1222 | An inference based on modus ponens. (Contributed by NM, 14-Jul-2005.) |
⊢ φ & ⊢ χ & ⊢ ((φ ∧ ψ ∧ χ) → θ) ⇒ ⊢ (ψ → θ) | ||
Theorem | mp3an23 1223 | An inference based on modus ponens. (Contributed by NM, 14-Jul-2005.) |
⊢ ψ & ⊢ χ & ⊢ ((φ ∧ ψ ∧ χ) → θ) ⇒ ⊢ (φ → θ) | ||
Theorem | mp3an1i 1224 | An inference based on modus ponens. (Contributed by NM, 5-Jul-2005.) |
⊢ ψ & ⊢ (φ → ((ψ ∧ χ ∧ θ) → τ)) ⇒ ⊢ (φ → ((χ ∧ θ) → τ)) | ||
Theorem | mp3anl1 1225 | An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.) |
⊢ φ & ⊢ (((φ ∧ ψ ∧ χ) ∧ θ) → τ) ⇒ ⊢ (((ψ ∧ χ) ∧ θ) → τ) | ||
Theorem | mp3anl2 1226 | An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.) |
⊢ ψ & ⊢ (((φ ∧ ψ ∧ χ) ∧ θ) → τ) ⇒ ⊢ (((φ ∧ χ) ∧ θ) → τ) | ||
Theorem | mp3anl3 1227 | An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.) |
⊢ χ & ⊢ (((φ ∧ ψ ∧ χ) ∧ θ) → τ) ⇒ ⊢ (((φ ∧ ψ) ∧ θ) → τ) | ||
Theorem | mp3anr1 1228 | An inference based on modus ponens. (Contributed by NM, 4-Nov-2006.) |
⊢ ψ & ⊢ ((φ ∧ (ψ ∧ χ ∧ θ)) → τ) ⇒ ⊢ ((φ ∧ (χ ∧ θ)) → τ) | ||
Theorem | mp3anr2 1229 | An inference based on modus ponens. (Contributed by NM, 24-Nov-2006.) |
⊢ χ & ⊢ ((φ ∧ (ψ ∧ χ ∧ θ)) → τ) ⇒ ⊢ ((φ ∧ (ψ ∧ θ)) → τ) | ||
Theorem | mp3anr3 1230 | An inference based on modus ponens. (Contributed by NM, 19-Oct-2007.) |
⊢ θ & ⊢ ((φ ∧ (ψ ∧ χ ∧ θ)) → τ) ⇒ ⊢ ((φ ∧ (ψ ∧ χ)) → τ) | ||
Theorem | mp3an 1231 | An inference based on modus ponens. (Contributed by NM, 14-May-1999.) |
⊢ φ & ⊢ ψ & ⊢ χ & ⊢ ((φ ∧ ψ ∧ χ) → θ) ⇒ ⊢ θ | ||
Theorem | mpd3an3 1232 | An inference based on modus ponens. (Contributed by NM, 8-Nov-2007.) |
⊢ ((φ ∧ ψ) → χ) & ⊢ ((φ ∧ ψ ∧ χ) → θ) ⇒ ⊢ ((φ ∧ ψ) → θ) | ||
Theorem | mpd3an23 1233 | An inference based on modus ponens. (Contributed by NM, 4-Dec-2006.) |
⊢ (φ → ψ) & ⊢ (φ → χ) & ⊢ ((φ ∧ ψ ∧ χ) → θ) ⇒ ⊢ (φ → θ) | ||
Theorem | biimp3a 1234 | Infer implication from a logical equivalence. Similar to biimpa 280. (Contributed by NM, 4-Sep-2005.) |
⊢ ((φ ∧ ψ) → (χ ↔ θ)) ⇒ ⊢ ((φ ∧ ψ ∧ χ) → θ) | ||
Theorem | biimp3ar 1235 | Infer implication from a logical equivalence. Similar to biimpar 281. (Contributed by NM, 2-Jan-2009.) |
⊢ ((φ ∧ ψ) → (χ ↔ θ)) ⇒ ⊢ ((φ ∧ ψ ∧ θ) → χ) | ||
Theorem | 3anandis 1236 | Inference that undistributes a triple conjunction in the antecedent. (Contributed by NM, 18-Apr-2007.) |
⊢ (((φ ∧ ψ) ∧ (φ ∧ χ) ∧ (φ ∧ θ)) → τ) ⇒ ⊢ ((φ ∧ (ψ ∧ χ ∧ θ)) → τ) | ||
Theorem | 3anandirs 1237 | Inference that undistributes a triple conjunction in the antecedent. (Contributed by NM, 25-Jul-2006.) (Revised by NM, 18-Apr-2007.) |
⊢ (((φ ∧ θ) ∧ (ψ ∧ θ) ∧ (χ ∧ θ)) → τ) ⇒ ⊢ (((φ ∧ ψ ∧ χ) ∧ θ) → τ) | ||
Theorem | ecased 1238 | Deduction form of disjunctive syllogism. (Contributed by Jim Kingdon, 9-Dec-2017.) |
⊢ (φ → ¬ χ) & ⊢ (φ → (ψ ∨ χ)) ⇒ ⊢ (φ → ψ) | ||
Theorem | ecase23d 1239 | Variation of ecased 1238 with three disjuncts instead of two. (Contributed by NM, 22-Apr-1994.) (Revised by Jim Kingdon, 9-Dec-2017.) |
⊢ (φ → ¬ χ) & ⊢ (φ → ¬ θ) & ⊢ (φ → (ψ ∨ χ ∨ θ)) ⇒ ⊢ (φ → ψ) | ||
Even though it isn't ordinarily part of propositional calculus, the universal quantifier ∀ is introduced here so that the soundness of definition df-tru 1245 can be checked by the same algorithm that is used for predicate calculus. Its first real use is in axiom ax-5 1333 in the predicate calculus section below. For those who want propositional calculus to be self-contained i.e. to use wff variables only, the alternate definition dftru2 1250 may be adopted and this subsection moved down to the start of the subsection with wex 1378 below. However, the use of dftru2 1250 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid. | ||
Syntax | wal 1240 | Extend wff definition to include the universal quantifier ('for all'). ∀xφ is read "φ (phi) is true for all x." Typically, in its final application φ would be replaced with a wff containing a (free) occurrence of the variable x, for example x = y. In a universe with a finite number of objects, "for all" is equivalent to a big conjunction (AND) with one wff for each possible case of x. When the universe is infinite (as with set theory), such a propositional-calculus equivalent is not possible because an infinitely long formula has no meaning, but conceptually the idea is the same. |
wff ∀xφ | ||
Even though it isn't ordinarily part of propositional calculus, the equality predicate = is introduced here so that the soundness of definition df-tru 1245 can be checked by the same algorithm as is used for predicate calculus. Its first real use is in axiom ax-8 1392 in the predicate calculus section below. For those who want propositional calculus to be self-contained i.e. to use wff variables only, the alternate definition dftru2 1250 may be adopted and this subsection moved down to just above weq 1389 below. However, the use of dftru2 1250 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid. | ||
Syntax | cv 1241 |
This syntax construction states that a variable x, which has been
declared to be a setvar variable by $f statement vx, is also a class
expression. This can be justified informally as follows. We know that
the class builder {y ∣ y
∈ x} is a class by cab 2023.
Since (when
y is
distinct from x) we
have x = {y ∣ y
∈ x} by
cvjust 2032, we can argue that the syntax "class x " can be viewed as
an abbreviation for "class
{y ∣ y ∈ x}". See the discussion
under the definition of class in [Jech] p.
4 showing that "Every set can
be considered to be a class."
While it is tempting and perhaps occasionally useful to view cv 1241 as a "type conversion" from a setvar variable to a class variable, keep in mind that cv 1241 is intrinsically no different from any other class-building syntax such as cab 2023, cun 2909, or c0 3218. For a general discussion of the theory of classes and the role of cv 1241, see http://us.metamath.org/mpeuni/mmset.html#class. (The description above applies to set theory, not predicate calculus. The purpose of introducing class x here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the weq 1389 of predicate calculus from the wceq 1242 of set theory, so that we don't overload the = connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers.) |
class x | ||
Syntax | wceq 1242 |
Extend wff definition to include class equality.
For a general discussion of the theory of classes, see http://us.metamath.org/mpeuni/mmset.html#class. (The purpose of introducing wff A = B here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the weq 1389 of predicate calculus in terms of the wceq 1242 of set theory, so that we don't "overload" the = connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers. For example, some parsers - although not the Metamath program - stumble on the fact that the = in x = y could be the = of either weq 1389 or wceq 1242, although mathematically it makes no difference. The class variables A and B are introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus. See df-cleq 2030 for more information on the set theory usage of wceq 1242.) |
wff A = B | ||
Syntax | wtru 1243 | ⊤ is a wff. |
wff ⊤ | ||
Theorem | trujust 1244 | Soundness justification theorem for df-tru 1245. (Contributed by Mario Carneiro, 17-Nov-2013.) (Revised by NM, 11-Jul-2019.) |
⊢ ((∀x x = x → ∀x x = x) ↔ (∀y y = y → ∀y y = y)) | ||
Definition | df-tru 1245 | Definition of the truth value "true", or "verum", denoted by ⊤. This is a tautology, as proved by tru 1246. In this definition, an instance of id 19 is used as the definiens, although any tautology, such as an axiom, can be used in its place. This particular id 19 instance was chosen so this definition can be checked by the same algorithm that is used for predicate calculus. This definition should be referenced directly only by tru 1246, and other proofs should depend on tru 1246 (directly or indirectly) instead of this definition, since there are many alternative ways to define ⊤. (Contributed by Anthony Hart, 13-Oct-2010.) (Revised by NM, 11-Jul-2019.) (New usage is discouraged.) |
⊢ ( ⊤ ↔ (∀x x = x → ∀x x = x)) | ||
Theorem | tru 1246 | The truth value ⊤ is provable. (Contributed by Anthony Hart, 13-Oct-2010.) |
⊢ ⊤ | ||
Syntax | wfal 1247 | ⊥ is a wff. |
wff ⊥ | ||
Definition | df-fal 1248 | Definition of the truth value "false", or "falsum", denoted by ⊥. See also df-tru 1245. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ ( ⊥ ↔ ¬ ⊤ ) | ||
Theorem | fal 1249 | The truth value ⊥ is refutable. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Mel L. O'Cat, 11-Mar-2012.) |
⊢ ¬ ⊥ | ||
Theorem | dftru2 1250 | An alternate definition of "true". (Contributed by Anthony Hart, 13-Oct-2010.) (Revised by BJ, 12-Jul-2019.) (New usage is discouraged.) |
⊢ ( ⊤ ↔ (φ → φ)) | ||
Theorem | trud 1251 | Eliminate ⊤ as an antecedent. A proposition implied by ⊤ is true. (Contributed by Mario Carneiro, 13-Mar-2014.) |
⊢ ( ⊤ → φ) ⇒ ⊢ φ | ||
Theorem | tbtru 1252 | A proposition is equivalent to itself being equivalent to ⊤. (Contributed by Anthony Hart, 14-Aug-2011.) |
⊢ (φ ↔ (φ ↔ ⊤ )) | ||
Theorem | nbfal 1253 | The negation of a proposition is equivalent to itself being equivalent to ⊥. (Contributed by Anthony Hart, 14-Aug-2011.) |
⊢ (¬ φ ↔ (φ ↔ ⊥ )) | ||
Theorem | bitru 1254 | A theorem is equivalent to truth. (Contributed by Mario Carneiro, 9-May-2015.) |
⊢ φ ⇒ ⊢ (φ ↔ ⊤ ) | ||
Theorem | bifal 1255 | A contradiction is equivalent to falsehood. (Contributed by Mario Carneiro, 9-May-2015.) |
⊢ ¬ φ ⇒ ⊢ (φ ↔ ⊥ ) | ||
Theorem | falim 1256 | The truth value ⊥ implies anything. Also called the principle of explosion, or "ex falso quodlibet". (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.) |
⊢ ( ⊥ → φ) | ||
Theorem | falimd 1257 | The truth value ⊥ implies anything. (Contributed by Mario Carneiro, 9-Feb-2017.) |
⊢ ((φ ∧ ⊥ ) → ψ) | ||
Theorem | a1tru 1258 | Anything implies ⊤. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.) |
⊢ (φ → ⊤ ) | ||
Theorem | truan 1259 | True can be removed from a conjunction. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Wolf Lammen, 21-Jul-2019.) |
⊢ (( ⊤ ∧ φ) ↔ φ) | ||
Theorem | truanOLD 1260 | Obsolete proof of truan 1259 as of 21-Jul-2019. (Contributed by FL, 20-Mar-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (( ⊤ ∧ φ) ↔ φ) | ||
Theorem | dfnot 1261 | Given falsum, we can define the negation of a wff φ as the statement that a contradiction follows from assuming φ. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof shortened by Wolf Lammen, 21-Jul-2019.) |
⊢ (¬ φ ↔ (φ → ⊥ )) | ||
Theorem | inegd 1262 | Negation introduction rule from natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.) |
⊢ ((φ ∧ ψ) → ⊥ ) ⇒ ⊢ (φ → ¬ ψ) | ||
Theorem | pm2.21fal 1263 | If a wff and its negation are provable, then falsum is provable. (Contributed by Mario Carneiro, 9-Feb-2017.) |
⊢ (φ → ψ) & ⊢ (φ → ¬ ψ) ⇒ ⊢ (φ → ⊥ ) | ||
Theorem | pclem6 1264 | Negation inferred from embedded conjunct. (Contributed by NM, 20-Aug-1993.) (Proof rewritten by Jim Kingdon, 4-May-2018.) |
⊢ ((φ ↔ (ψ ∧ ¬ φ)) → ¬ ψ) | ||
Syntax | wxo 1265 | Extend wff definition to include exclusive disjunction ('xor'). |
wff (φ ⊻ ψ) | ||
Definition | df-xor 1266 | Define exclusive disjunction (logical 'xor'). Return true if either the left or right, but not both, are true. Contrast with ∧ (wa 97), ∨ (wo 628), and → (wi 4) . (Contributed by FL, 22-Nov-2010.) (Modified by Jim Kingdon, 1-Mar-2018.) |
⊢ ((φ ⊻ ψ) ↔ ((φ ∨ ψ) ∧ ¬ (φ ∧ ψ))) | ||
Theorem | xoranor 1267 | One way of defining exclusive or. Equivalent to df-xor 1266. (Contributed by Jim Kingdon and Mario Carneiro, 1-Mar-2018.) |
⊢ ((φ ⊻ ψ) ↔ ((φ ∨ ψ) ∧ (¬ φ ∨ ¬ ψ))) | ||
Theorem | excxor 1268 | This tautology shows that xor is really exclusive. (Contributed by FL, 22-Nov-2010.) (Proof rewritten by Jim Kingdon, 5-May-2018.) |
⊢ ((φ ⊻ ψ) ↔ ((φ ∧ ¬ ψ) ∨ (¬ φ ∧ ψ))) | ||
Theorem | xorbi2d 1269 | Deduction joining an equivalence and a left operand to form equivalence of exclusive-or. (Contributed by Jim Kingdon, 7-Oct-2018.) |
⊢ (φ → (ψ ↔ χ)) ⇒ ⊢ (φ → ((θ ⊻ ψ) ↔ (θ ⊻ χ))) | ||
Theorem | xorbi1d 1270 | Deduction joining an equivalence and a right operand to form equivalence of exclusive-or. (Contributed by Jim Kingdon, 7-Oct-2018.) |
⊢ (φ → (ψ ↔ χ)) ⇒ ⊢ (φ → ((ψ ⊻ θ) ↔ (χ ⊻ θ))) | ||
Theorem | xorbi12d 1271 | Deduction joining two equivalences to form equivalence of exclusive-or. (Contributed by Jim Kingdon, 7-Oct-2018.) |
⊢ (φ → (ψ ↔ χ)) & ⊢ (φ → (θ ↔ τ)) ⇒ ⊢ (φ → ((ψ ⊻ θ) ↔ (χ ⊻ τ))) | ||
Theorem | xorbin 1272 | A consequence of exclusive or. In classical logic the converse also holds. (Contributed by Jim Kingdon, 8-Mar-2018.) |
⊢ ((φ ⊻ ψ) → (φ ↔ ¬ ψ)) | ||
Theorem | pm5.18im 1273 | One direction of pm5.18dc 776, which holds for all propositions, not just decidable propositions. (Contributed by Jim Kingdon, 10-Mar-2018.) |
⊢ ((φ ↔ ψ) → ¬ (φ ↔ ¬ ψ)) | ||
Theorem | xornbi 1274 | A consequence of exclusive or. For decidable propositions this is an equivalence, as seen at xornbidc 1279. (Contributed by Jim Kingdon, 10-Mar-2018.) |
⊢ ((φ ⊻ ψ) → ¬ (φ ↔ ψ)) | ||
Theorem | xor3dc 1275 | Two ways to express "exclusive or" between decidable propositions. (Contributed by Jim Kingdon, 12-Apr-2018.) |
⊢ (DECID φ → (DECID ψ → (¬ (φ ↔ ψ) ↔ (φ ↔ ¬ ψ)))) | ||
Theorem | xorcom 1276 | ⊻ is commutative. (Contributed by David A. Wheeler, 6-Oct-2018.) |
⊢ ((φ ⊻ ψ) ↔ (ψ ⊻ φ)) | ||
Theorem | pm5.15dc 1277 | A decidable proposition is equivalent to a decidable proposition or its negation. Based on theorem *5.15 of [WhiteheadRussell] p. 124. (Contributed by Jim Kingdon, 18-Apr-2018.) |
⊢ (DECID φ → (DECID ψ → ((φ ↔ ψ) ∨ (φ ↔ ¬ ψ)))) | ||
Theorem | xor2dc 1278 | Two ways to express "exclusive or" between decidable propositions. (Contributed by Jim Kingdon, 17-Apr-2018.) |
⊢ (DECID φ → (DECID ψ → (¬ (φ ↔ ψ) ↔ ((φ ∨ ψ) ∧ ¬ (φ ∧ ψ))))) | ||
Theorem | xornbidc 1279 | Exclusive or is equivalent to negated biconditional for decidable propositions. (Contributed by Jim Kingdon, 27-Apr-2018.) |
⊢ (DECID φ → (DECID ψ → ((φ ⊻ ψ) ↔ ¬ (φ ↔ ψ)))) | ||
Theorem | xordc 1280 | Two ways to express "exclusive or" between decidable propositions. Theorem *5.22 of [WhiteheadRussell] p. 124, but for decidable propositions. (Contributed by Jim Kingdon, 5-May-2018.) |
⊢ (DECID φ → (DECID ψ → (¬ (φ ↔ ψ) ↔ ((φ ∧ ¬ ψ) ∨ (ψ ∧ ¬ φ))))) | ||
Theorem | xordc1 1281 | Exclusive or implies the left proposition is decidable. (Contributed by Jim Kingdon, 12-Mar-2018.) |
⊢ ((φ ⊻ ψ) → DECID φ) | ||
Theorem | nbbndc 1282 | Move negation outside of biconditional, for decidable propositions. Compare Theorem *5.18 of [WhiteheadRussell] p. 124. (Contributed by Jim Kingdon, 18-Apr-2018.) |
⊢ (DECID φ → (DECID ψ → ((¬ φ ↔ ψ) ↔ ¬ (φ ↔ ψ)))) | ||
Theorem | biassdc 1283 |
Associative law for the biconditional, for decidable propositions.
The classical version (without the decidability conditions) is an axiom of system DS in Vladimir Lifschitz, "On calculational proofs", Annals of Pure and Applied Logic, 113:207-224, 2002, http://www.cs.utexas.edu/users/ai-lab/pub-view.php?PubID=26805, and, interestingly, was not included in Principia Mathematica but was apparently first noted by Jan Lukasiewicz circa 1923. (Contributed by Jim Kingdon, 4-May-2018.) |
⊢ (DECID φ → (DECID ψ → (DECID χ → (((φ ↔ ψ) ↔ χ) ↔ (φ ↔ (ψ ↔ χ)))))) | ||
Theorem | bilukdc 1284 | Lukasiewicz's shortest axiom for equivalential calculus (but modified to require decidable propositions). Storrs McCall, ed., Polish Logic 1920-1939 (Oxford, 1967), p. 96. (Contributed by Jim Kingdon, 5-May-2018.) |
⊢ (((DECID φ ∧ DECID ψ) ∧ DECID χ) → ((φ ↔ ψ) ↔ ((χ ↔ ψ) ↔ (φ ↔ χ)))) | ||
Theorem | dfbi3dc 1285 | An alternate definition of the biconditional for decidable propositions. Theorem *5.23 of [WhiteheadRussell] p. 124, but with decidability conditions. (Contributed by Jim Kingdon, 5-May-2018.) |
⊢ (DECID φ → (DECID ψ → ((φ ↔ ψ) ↔ ((φ ∧ ψ) ∨ (¬ φ ∧ ¬ ψ))))) | ||
Theorem | pm5.24dc 1286 | Theorem *5.24 of [WhiteheadRussell] p. 124, but for decidable propositions. (Contributed by Jim Kingdon, 5-May-2018.) |
⊢ (DECID φ → (DECID ψ → (¬ ((φ ∧ ψ) ∨ (¬ φ ∧ ¬ ψ)) ↔ ((φ ∧ ¬ ψ) ∨ (ψ ∧ ¬ φ))))) | ||
Theorem | xordidc 1287 | Conjunction distributes over exclusive-or, for decidable propositions. This is one way to interpret the distributive law of multiplication over addition in modulo 2 arithmetic. (Contributed by Jim Kingdon, 14-Jul-2018.) |
⊢ (DECID φ → (DECID ψ → (DECID χ → ((φ ∧ (ψ ⊻ χ)) ↔ ((φ ∧ ψ) ⊻ (φ ∧ χ)))))) | ||
Theorem | anxordi 1288 | Conjunction distributes over exclusive-or. (Contributed by Mario Carneiro and Jim Kingdon, 7-Oct-2018.) |
⊢ ((φ ∧ (ψ ⊻ χ)) ↔ ((φ ∧ ψ) ⊻ (φ ∧ χ))) | ||
For classical logic, truth tables can be used to define propositional logic operations, by showing the results of those operations for all possible combinations of true (⊤) and false (⊥). Although the intuitionistic logic connectives are not as simply defined, ⊤ and ⊥ do play similar roles as in classical logic and most theorems from classical logic continue to hold. Here we show that our definitions and axioms produce equivalent results for ⊤ and ⊥ as we would get from truth tables for ∧ (conjunction aka logical 'and') wa 97, ∨ (disjunction aka logical inclusive 'or') wo 628, → (implies) wi 4, ¬ (not) wn 3, ↔ (logical equivalence) df-bi 110, and ⊻ (exclusive or) df-xor 1266. | ||
Theorem | truantru 1289 | A ∧ identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ (( ⊤ ∧ ⊤ ) ↔ ⊤ ) | ||
Theorem | truanfal 1290 | A ∧ identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ (( ⊤ ∧ ⊥ ) ↔ ⊥ ) | ||
Theorem | falantru 1291 | A ∧ identity. (Contributed by David A. Wheeler, 23-Feb-2018.) |
⊢ (( ⊥ ∧ ⊤ ) ↔ ⊥ ) | ||
Theorem | falanfal 1292 | A ∧ identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ (( ⊥ ∧ ⊥ ) ↔ ⊥ ) | ||
Theorem | truortru 1293 | A ∨ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
⊢ (( ⊤ ∨ ⊤ ) ↔ ⊤ ) | ||
Theorem | truorfal 1294 | A ∨ identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ (( ⊤ ∨ ⊥ ) ↔ ⊤ ) | ||
Theorem | falortru 1295 | A ∨ identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ (( ⊥ ∨ ⊤ ) ↔ ⊤ ) | ||
Theorem | falorfal 1296 | A ∨ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
⊢ (( ⊥ ∨ ⊥ ) ↔ ⊥ ) | ||
Theorem | truimtru 1297 | A → identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ (( ⊤ → ⊤ ) ↔ ⊤ ) | ||
Theorem | truimfal 1298 | A → identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
⊢ (( ⊤ → ⊥ ) ↔ ⊥ ) | ||
Theorem | falimtru 1299 | A → identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ (( ⊥ → ⊤ ) ↔ ⊤ ) | ||
Theorem | falimfal 1300 | A → identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ (( ⊥ → ⊥ ) ↔ ⊤ ) |
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