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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | 3an6 1301 | Analog of an4 576 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 1302 | Analog of or4 761 for triple conjunction. (Contributed by Scott Fenton, 16-Mar-2011.) |
⊢ (((𝜑 ∨ 𝜓) ∨ (𝜒 ∨ 𝜃) ∨ (𝜏 ∨ 𝜂)) ↔ ((𝜑 ∨ 𝜒 ∨ 𝜏) ∨ (𝜓 ∨ 𝜃 ∨ 𝜂))) | ||
Theorem | mp3an1 1303 | An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.) |
⊢ 𝜑 & ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜓 ∧ 𝜒) → 𝜃) | ||
Theorem | mp3an2 1304 | An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.) |
⊢ 𝜓 & ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜒) → 𝜃) | ||
Theorem | mp3an3 1305 | An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.) |
⊢ 𝜒 & ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜃) | ||
Theorem | mp3an12 1306 | An inference based on modus ponens. (Contributed by NM, 13-Jul-2005.) |
⊢ 𝜑 & ⊢ 𝜓 & ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ (𝜒 → 𝜃) | ||
Theorem | mp3an13 1307 | An inference based on modus ponens. (Contributed by NM, 14-Jul-2005.) |
⊢ 𝜑 & ⊢ 𝜒 & ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ (𝜓 → 𝜃) | ||
Theorem | mp3an23 1308 | An inference based on modus ponens. (Contributed by NM, 14-Jul-2005.) |
⊢ 𝜓 & ⊢ 𝜒 & ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ (𝜑 → 𝜃) | ||
Theorem | mp3an1i 1309 | An inference based on modus ponens. (Contributed by NM, 5-Jul-2005.) |
⊢ 𝜓 & ⊢ (𝜑 → ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)) ⇒ ⊢ (𝜑 → ((𝜒 ∧ 𝜃) → 𝜏)) | ||
Theorem | mp3anl1 1310 | An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.) |
⊢ 𝜑 & ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏) ⇒ ⊢ (((𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏) | ||
Theorem | mp3anl2 1311 | An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.) |
⊢ 𝜓 & ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏) ⇒ ⊢ (((𝜑 ∧ 𝜒) ∧ 𝜃) → 𝜏) | ||
Theorem | mp3anl3 1312 | An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.) |
⊢ 𝜒 & ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏) ⇒ ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜃) → 𝜏) | ||
Theorem | mp3anr1 1313 | An inference based on modus ponens. (Contributed by NM, 4-Nov-2006.) |
⊢ 𝜓 & ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) → 𝜏) ⇒ ⊢ ((𝜑 ∧ (𝜒 ∧ 𝜃)) → 𝜏) | ||
Theorem | mp3anr2 1314 | An inference based on modus ponens. (Contributed by NM, 24-Nov-2006.) |
⊢ 𝜒 & ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) → 𝜏) ⇒ ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜃)) → 𝜏) | ||
Theorem | mp3anr3 1315 | An inference based on modus ponens. (Contributed by NM, 19-Oct-2007.) |
⊢ 𝜃 & ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) → 𝜏) ⇒ ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜏) | ||
Theorem | mp3an 1316 | An inference based on modus ponens. (Contributed by NM, 14-May-1999.) |
⊢ 𝜑 & ⊢ 𝜓 & ⊢ 𝜒 & ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ 𝜃 | ||
Theorem | mpd3an3 1317 | An inference based on modus ponens. (Contributed by NM, 8-Nov-2007.) |
⊢ ((𝜑 ∧ 𝜓) → 𝜒) & ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜃) | ||
Theorem | mpd3an23 1318 | An inference based on modus ponens. (Contributed by NM, 4-Dec-2006.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜑 → 𝜒) & ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ (𝜑 → 𝜃) | ||
Theorem | mp3and 1319 | A deduction based on modus ponens. (Contributed by Mario Carneiro, 24-Dec-2016.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜑 → 𝜒) & ⊢ (𝜑 → 𝜃) & ⊢ (𝜑 → ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏)) ⇒ ⊢ (𝜑 → 𝜏) | ||
Theorem | mp3an12i 1320 | mp3an 1316 with antecedents in standard conjunction form and with one hypothesis an implication. (Contributed by Alan Sare, 28-Aug-2016.) |
⊢ 𝜑 & ⊢ 𝜓 & ⊢ (𝜒 → 𝜃) & ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏) ⇒ ⊢ (𝜒 → 𝜏) | ||
Theorem | mp3an2i 1321 | mp3an 1316 with antecedents in standard conjunction form and with two hypotheses which are implications. (Contributed by Alan Sare, 28-Aug-2016.) |
⊢ 𝜑 & ⊢ (𝜓 → 𝜒) & ⊢ (𝜓 → 𝜃) & ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜏) ⇒ ⊢ (𝜓 → 𝜏) | ||
Theorem | mp3an3an 1322 | mp3an 1316 with antecedents in standard conjunction form and with two hypotheses which are implications. (Contributed by Alan Sare, 28-Aug-2016.) |
⊢ 𝜑 & ⊢ (𝜓 → 𝜒) & ⊢ (𝜃 → 𝜏) & ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜂) ⇒ ⊢ ((𝜓 ∧ 𝜃) → 𝜂) | ||
Theorem | mp3an2ani 1323 | An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.) |
⊢ 𝜑 & ⊢ (𝜓 → 𝜒) & ⊢ ((𝜓 ∧ 𝜃) → 𝜏) & ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜂) ⇒ ⊢ ((𝜓 ∧ 𝜃) → 𝜂) | ||
Theorem | biimp3a 1324 | Infer implication from a logical equivalence. Similar to biimpa 294. (Contributed by NM, 4-Sep-2005.) |
⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | ||
Theorem | biimp3ar 1325 | Infer implication from a logical equivalence. Similar to biimpar 295. (Contributed by NM, 2-Jan-2009.) |
⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜒) | ||
Theorem | 3anandis 1326 | 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 1327 | 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 1328 | Deduction form of disjunctive syllogism. (Contributed by Jim Kingdon, 9-Dec-2017.) |
⊢ (𝜑 → ¬ 𝜒) & ⊢ (𝜑 → (𝜓 ∨ 𝜒)) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | ecase23d 1329 | Variation of ecased 1328 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 is not ordinarily part of propositional calculus, the universal quantifier ∀ is introduced here so that the soundness of Definition df-tru 1335 can be checked by the same algorithm that is used for predicate calculus. Its first real use is in Axiom ax-5 1424 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 1340 may be adopted and this subsection moved down to the start of the subsection with wex 1469 below. However, the use of dftru2 1340 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid. | ||
Syntax | wal 1330 | 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 is not ordinarily part of propositional calculus, the equality predicate = is introduced here so that the soundness of definition df-tru 1335 can be checked by the same algorithm as is used for predicate calculus. Its first real use is in Axiom ax-8 1481 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 1340 may be adopted and this subsection moved down to just above weq 1480 below. However, the use of dftru2 1340 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid. | ||
Syntax | cv 1331 |
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 2140.
Since (when
𝑦 is distinct from 𝑥) we
have 𝑥 =
{𝑦 ∣ 𝑦 ∈ 𝑥} by
cvjust 2149, 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 1331 as a "type conversion" from a setvar variable to a class variable, keep in mind that cv 1331 is intrinsically no different from any other class-building syntax such as cab 2140, cun 3096, or c0 3390. For a general discussion of the theory of classes and the role of cv 1331, see https://us.metamath.org/mpeuni/mmset.html#class 1331. (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 1480 of predicate calculus from the wceq 1332 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 1332 |
Extend wff definition to include class equality.
For a general discussion of the theory of classes, see https://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 1480 of predicate calculus in terms of the wceq 1332 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 1480 or wceq 1332, 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 2147 for more information on the set theory usage of wceq 1332.) |
wff 𝐴 = 𝐵 | ||
Syntax | wtru 1333 | ⊤ is a wff. |
wff ⊤ | ||
Theorem | trujust 1334 | Soundness justification theorem for df-tru 1335. (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 1335 | Definition of the truth value "true", or "verum", denoted by ⊤. This is a tautology, as proved by tru 1336. 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 1336, and other proofs should depend on tru 1336 (directly or indirectly) instead of this definition, since there are many alternate 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 1336 | The truth value ⊤ is provable. (Contributed by Anthony Hart, 13-Oct-2010.) |
⊢ ⊤ | ||
Syntax | wfal 1337 | ⊥ is a wff. |
wff ⊥ | ||
Definition | df-fal 1338 | Definition of the truth value "false", or "falsum", denoted by ⊥. See also df-tru 1335. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ (⊥ ↔ ¬ ⊤) | ||
Theorem | fal 1339 | The truth value ⊥ is refutable. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Mel L. O'Cat, 11-Mar-2012.) |
⊢ ¬ ⊥ | ||
Theorem | dftru2 1340 | 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 | mptru 1341 | Eliminate ⊤ as an antecedent. A proposition implied by ⊤ is true. (Contributed by Mario Carneiro, 13-Mar-2014.) |
⊢ (⊤ → 𝜑) ⇒ ⊢ 𝜑 | ||
Theorem | tbtru 1342 | A proposition is equivalent to itself being equivalent to ⊤. (Contributed by Anthony Hart, 14-Aug-2011.) |
⊢ (𝜑 ↔ (𝜑 ↔ ⊤)) | ||
Theorem | nbfal 1343 | The negation of a proposition is equivalent to itself being equivalent to ⊥. (Contributed by Anthony Hart, 14-Aug-2011.) |
⊢ (¬ 𝜑 ↔ (𝜑 ↔ ⊥)) | ||
Theorem | bitru 1344 | A theorem is equivalent to truth. (Contributed by Mario Carneiro, 9-May-2015.) |
⊢ 𝜑 ⇒ ⊢ (𝜑 ↔ ⊤) | ||
Theorem | bifal 1345 | A contradiction is equivalent to falsehood. (Contributed by Mario Carneiro, 9-May-2015.) |
⊢ ¬ 𝜑 ⇒ ⊢ (𝜑 ↔ ⊥) | ||
Theorem | falim 1346 | 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 1347 | The truth value ⊥ implies anything. (Contributed by Mario Carneiro, 9-Feb-2017.) |
⊢ ((𝜑 ∧ ⊥) → 𝜓) | ||
Theorem | a1tru 1348 | Anything implies ⊤. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.) |
⊢ (𝜑 → ⊤) | ||
Theorem | truan 1349 | True can be removed from a conjunction. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Wolf Lammen, 21-Jul-2019.) |
⊢ ((⊤ ∧ 𝜑) ↔ 𝜑) | ||
Theorem | dfnot 1350 | 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 1351 | Negation introduction rule from natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.) |
⊢ ((𝜑 ∧ 𝜓) → ⊥) ⇒ ⊢ (𝜑 → ¬ 𝜓) | ||
Theorem | pm2.21fal 1352 | 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 1353 | 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 1354 | Extend wff definition to include exclusive disjunction ('xor'). |
wff (𝜑 ⊻ 𝜓) | ||
Definition | df-xor 1355 | Define exclusive disjunction (logical 'xor'). Return true if either the left or right, but not both, are true. Contrast with ∧ (wa 103), ∨ (wo 698), and → (wi 4) . (Contributed by FL, 22-Nov-2010.) (Modified by Jim Kingdon, 1-Mar-2018.) |
⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓))) | ||
Theorem | xoranor 1356 | One way of defining exclusive or. Equivalent to df-xor 1355. (Contributed by Jim Kingdon and Mario Carneiro, 1-Mar-2018.) |
⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ (¬ 𝜑 ∨ ¬ 𝜓))) | ||
Theorem | excxor 1357 | 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 | xoror 1358 | XOR implies OR. (Contributed by BJ, 19-Apr-2019.) |
⊢ ((𝜑 ⊻ 𝜓) → (𝜑 ∨ 𝜓)) | ||
Theorem | xorbi2d 1359 | 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 1360 | 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 1361 | 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 | xorbi12i 1362 | Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.) |
⊢ (𝜑 ↔ 𝜓) & ⊢ (𝜒 ↔ 𝜃) ⇒ ⊢ ((𝜑 ⊻ 𝜒) ↔ (𝜓 ⊻ 𝜃)) | ||
Theorem | xorbin 1363 | 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 1364 | One direction of pm5.18dc 869, which holds for all propositions, not just decidable propositions. (Contributed by Jim Kingdon, 10-Mar-2018.) |
⊢ ((𝜑 ↔ 𝜓) → ¬ (𝜑 ↔ ¬ 𝜓)) | ||
Theorem | xornbi 1365 | A consequence of exclusive or. For decidable propositions this is an equivalence, as seen at xornbidc 1370. (Contributed by Jim Kingdon, 10-Mar-2018.) |
⊢ ((𝜑 ⊻ 𝜓) → ¬ (𝜑 ↔ 𝜓)) | ||
Theorem | xor3dc 1366 | 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 1367 | ⊻ is commutative. (Contributed by David A. Wheeler, 6-Oct-2018.) |
⊢ ((𝜑 ⊻ 𝜓) ↔ (𝜓 ⊻ 𝜑)) | ||
Theorem | pm5.15dc 1368 | 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 1369 | 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 1370 | 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 1371 | 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 1372 | Exclusive or implies the left proposition is decidable. (Contributed by Jim Kingdon, 12-Mar-2018.) |
⊢ ((𝜑 ⊻ 𝜓) → DECID 𝜑) | ||
Theorem | nbbndc 1373 | 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 1374 |
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 1375 | 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 1376 | 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 1377 | 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 1378 | 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 1379 | 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 103, ∨ (disjunction aka logical inclusive 'or') wo 698, → (implies) wi 4, ¬ (not) wn 3, ↔ (logical equivalence) df-bi 116, and ⊻ (exclusive or) df-xor 1355. | ||
Theorem | truantru 1380 | A ∧ identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ ((⊤ ∧ ⊤) ↔ ⊤) | ||
Theorem | truanfal 1381 | A ∧ identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ ((⊤ ∧ ⊥) ↔ ⊥) | ||
Theorem | falantru 1382 | A ∧ identity. (Contributed by David A. Wheeler, 23-Feb-2018.) |
⊢ ((⊥ ∧ ⊤) ↔ ⊥) | ||
Theorem | falanfal 1383 | A ∧ identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ ((⊥ ∧ ⊥) ↔ ⊥) | ||
Theorem | truortru 1384 | A ∨ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
⊢ ((⊤ ∨ ⊤) ↔ ⊤) | ||
Theorem | truorfal 1385 | A ∨ identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ ((⊤ ∨ ⊥) ↔ ⊤) | ||
Theorem | falortru 1386 | A ∨ identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ ((⊥ ∨ ⊤) ↔ ⊤) | ||
Theorem | falorfal 1387 | A ∨ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
⊢ ((⊥ ∨ ⊥) ↔ ⊥) | ||
Theorem | truimtru 1388 | A → identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ ((⊤ → ⊤) ↔ ⊤) | ||
Theorem | truimfal 1389 | A → identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
⊢ ((⊤ → ⊥) ↔ ⊥) | ||
Theorem | falimtru 1390 | A → identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ ((⊥ → ⊤) ↔ ⊤) | ||
Theorem | falimfal 1391 | A → identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ ((⊥ → ⊥) ↔ ⊤) | ||
Theorem | nottru 1392 | A ¬ identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ (¬ ⊤ ↔ ⊥) | ||
Theorem | notfal 1393 | A ¬ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
⊢ (¬ ⊥ ↔ ⊤) | ||
Theorem | trubitru 1394 | A ↔ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
⊢ ((⊤ ↔ ⊤) ↔ ⊤) | ||
Theorem | trubifal 1395 | A ↔ identity. (Contributed by David A. Wheeler, 23-Feb-2018.) |
⊢ ((⊤ ↔ ⊥) ↔ ⊥) | ||
Theorem | falbitru 1396 | A ↔ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
⊢ ((⊥ ↔ ⊤) ↔ ⊥) | ||
Theorem | falbifal 1397 | A ↔ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
⊢ ((⊥ ↔ ⊥) ↔ ⊤) | ||
Theorem | truxortru 1398 | A ⊻ identity. (Contributed by David A. Wheeler, 2-Mar-2018.) |
⊢ ((⊤ ⊻ ⊤) ↔ ⊥) | ||
Theorem | truxorfal 1399 | A ⊻ identity. (Contributed by David A. Wheeler, 2-Mar-2018.) |
⊢ ((⊤ ⊻ ⊥) ↔ ⊤) | ||
Theorem | falxortru 1400 | A ⊻ identity. (Contributed by David A. Wheeler, 2-Mar-2018.) |
⊢ ((⊥ ⊻ ⊤) ↔ ⊤) |
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