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Theorem List for Metamath Proof Explorer - 1401-1500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem3or6 1401 Analogue of or4 548 for triple conjunction. (Contributed by Scott Fenton, 16-Mar-2011.)
(((𝜑𝜓) ∨ (𝜒𝜃) ∨ (𝜏𝜂)) ↔ ((𝜑𝜒𝜏) ∨ (𝜓𝜃𝜂)))
 
Theoremmp3an1 1402 An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.)
𝜑    &   ((𝜑𝜓𝜒) → 𝜃)       ((𝜓𝜒) → 𝜃)
 
Theoremmp3an2 1403 An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.)
𝜓    &   ((𝜑𝜓𝜒) → 𝜃)       ((𝜑𝜒) → 𝜃)
 
Theoremmp3an3 1404 An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.)
𝜒    &   ((𝜑𝜓𝜒) → 𝜃)       ((𝜑𝜓) → 𝜃)
 
Theoremmp3an12 1405 An inference based on modus ponens. (Contributed by NM, 13-Jul-2005.)
𝜑    &   𝜓    &   ((𝜑𝜓𝜒) → 𝜃)       (𝜒𝜃)
 
Theoremmp3an13 1406 An inference based on modus ponens. (Contributed by NM, 14-Jul-2005.)
𝜑    &   𝜒    &   ((𝜑𝜓𝜒) → 𝜃)       (𝜓𝜃)
 
Theoremmp3an23 1407 An inference based on modus ponens. (Contributed by NM, 14-Jul-2005.)
𝜓    &   𝜒    &   ((𝜑𝜓𝜒) → 𝜃)       (𝜑𝜃)
 
Theoremmp3an1i 1408 An inference based on modus ponens. (Contributed by NM, 5-Jul-2005.)
𝜓    &   (𝜑 → ((𝜓𝜒𝜃) → 𝜏))       (𝜑 → ((𝜒𝜃) → 𝜏))
 
Theoremmp3anl1 1409 An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.)
𝜑    &   (((𝜑𝜓𝜒) ∧ 𝜃) → 𝜏)       (((𝜓𝜒) ∧ 𝜃) → 𝜏)
 
Theoremmp3anl2 1410 An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.)
𝜓    &   (((𝜑𝜓𝜒) ∧ 𝜃) → 𝜏)       (((𝜑𝜒) ∧ 𝜃) → 𝜏)
 
Theoremmp3anl3 1411 An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.)
𝜒    &   (((𝜑𝜓𝜒) ∧ 𝜃) → 𝜏)       (((𝜑𝜓) ∧ 𝜃) → 𝜏)
 
Theoremmp3anr1 1412 An inference based on modus ponens. (Contributed by NM, 4-Nov-2006.)
𝜓    &   ((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜏)       ((𝜑 ∧ (𝜒𝜃)) → 𝜏)
 
Theoremmp3anr2 1413 An inference based on modus ponens. (Contributed by NM, 24-Nov-2006.)
𝜒    &   ((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜏)       ((𝜑 ∧ (𝜓𝜃)) → 𝜏)
 
Theoremmp3anr3 1414 An inference based on modus ponens. (Contributed by NM, 19-Oct-2007.)
𝜃    &   ((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜏)       ((𝜑 ∧ (𝜓𝜒)) → 𝜏)
 
Theoremmp3an 1415 An inference based on modus ponens. (Contributed by NM, 14-May-1999.)
𝜑    &   𝜓    &   𝜒    &   ((𝜑𝜓𝜒) → 𝜃)       𝜃
 
Theoremmpd3an3 1416 An inference based on modus ponens. (Contributed by NM, 8-Nov-2007.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜓𝜒) → 𝜃)       ((𝜑𝜓) → 𝜃)
 
Theoremmpd3an23 1417 An inference based on modus ponens. (Contributed by NM, 4-Dec-2006.)
(𝜑𝜓)    &   (𝜑𝜒)    &   ((𝜑𝜓𝜒) → 𝜃)       (𝜑𝜃)
 
Theoremmp3and 1418 A deduction based on modus ponens. (Contributed by Mario Carneiro, 24-Dec-2016.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑 → ((𝜓𝜒𝜃) → 𝜏))       (𝜑𝜏)
 
Theoremmp3an12i 1419 mp3an 1415 with antecedents in standard conjunction form and with one hypothesis an implication. (Contributed by Alan Sare, 28-Aug-2016.)
𝜑    &   𝜓    &   (𝜒𝜃)    &   ((𝜑𝜓𝜃) → 𝜏)       (𝜒𝜏)
 
Theoremmp3an2i 1420 mp3an 1415 with antecedents in standard conjunction form and with two hypotheses which are implications. (Contributed by Alan Sare, 28-Aug-2016.)
𝜑    &   (𝜓𝜒)    &   (𝜓𝜃)    &   ((𝜑𝜒𝜃) → 𝜏)       (𝜓𝜏)
 
Theoremmp3an3an 1421 mp3an 1415 with antecedents in standard conjunction form and with two hypotheses which are implications. (Contributed by Alan Sare, 28-Aug-2016.)
𝜑    &   (𝜓𝜒)    &   (𝜃𝜏)    &   ((𝜑𝜒𝜏) → 𝜂)       ((𝜓𝜃) → 𝜂)
 
Theoremmp3an2ani 1422 An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.)
𝜑    &   (𝜓𝜒)    &   ((𝜓𝜃) → 𝜏)    &   ((𝜑𝜒𝜏) → 𝜂)       ((𝜓𝜃) → 𝜂)
 
Theorembiimp3a 1423 Infer implication from a logical equivalence. Similar to biimpa 499. (Contributed by NM, 4-Sep-2005.)
((𝜑𝜓) → (𝜒𝜃))       ((𝜑𝜓𝜒) → 𝜃)
 
Theorembiimp3ar 1424 Infer implication from a logical equivalence. Similar to biimpar 500. (Contributed by NM, 2-Jan-2009.)
((𝜑𝜓) → (𝜒𝜃))       ((𝜑𝜓𝜃) → 𝜒)
 
Theorem3anandis 1425 Inference that undistributes a triple conjunction in the antecedent. (Contributed by NM, 18-Apr-2007.)
(((𝜑𝜓) ∧ (𝜑𝜒) ∧ (𝜑𝜃)) → 𝜏)       ((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜏)
 
Theorem3anandirs 1426 Inference that undistributes a triple conjunction in the antecedent. (Contributed by NM, 25-Jul-2006.)
(((𝜑𝜃) ∧ (𝜓𝜃) ∧ (𝜒𝜃)) → 𝜏)       (((𝜑𝜓𝜒) ∧ 𝜃) → 𝜏)
 
Theoremecase23d 1427 Deduction for elimination by cases. (Contributed by NM, 22-Apr-1994.)
(𝜑 → ¬ 𝜒)    &   (𝜑 → ¬ 𝜃)    &   (𝜑 → (𝜓𝜒𝜃))       (𝜑𝜓)
 
Theorem3ecase 1428 Inference for elimination by cases. (Contributed by NM, 13-Jul-2005.)
𝜑𝜃)    &   𝜓𝜃)    &   𝜒𝜃)    &   ((𝜑𝜓𝜒) → 𝜃)       𝜃
 
Theorem3bior1fd 1429 A disjunction is equivalent to a threefold disjunction with single falsehood, analogous to biorf 418. (Contributed by Alexander van der Vekens, 8-Sep-2017.)
(𝜑 → ¬ 𝜃)       (𝜑 → ((𝜒𝜓) ↔ (𝜃𝜒𝜓)))
 
Theorem3bior1fand 1430 A disjunction is equivalent to a threefold disjunction with single falsehood of a conjunction. (Contributed by Alexander van der Vekens, 8-Sep-2017.)
(𝜑 → ¬ 𝜃)       (𝜑 → ((𝜒𝜓) ↔ ((𝜃𝜏) ∨ 𝜒𝜓)))
 
Theorem3bior2fd 1431 A wff is equivalent to its threefold disjunction with double falsehood, analogous to biorf 418. (Contributed by Alexander van der Vekens, 8-Sep-2017.)
(𝜑 → ¬ 𝜃)    &   (𝜑 → ¬ 𝜒)       (𝜑 → (𝜓 ↔ (𝜃𝜒𝜓)))
 
Theorem3biant1d 1432 A conjunction is equivalent to a threefold conjunction with single truth, analogous to biantrud 526. (Contributed by Alexander van der Vekens, 26-Sep-2017.)
(𝜑𝜃)       (𝜑 → ((𝜒𝜓) ↔ (𝜃𝜒𝜓)))
 
Theoremintn3an1d 1433 Introduction of a triple conjunct inside a contradiction. (Contributed by FL, 27-Dec-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝜑 → ¬ 𝜓)       (𝜑 → ¬ (𝜓𝜒𝜃))
 
Theoremintn3an2d 1434 Introduction of a triple conjunct inside a contradiction. (Contributed by FL, 27-Dec-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝜑 → ¬ 𝜓)       (𝜑 → ¬ (𝜒𝜓𝜃))
 
Theoremintn3an3d 1435 Introduction of a triple conjunct inside a contradiction. (Contributed by FL, 27-Dec-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝜑 → ¬ 𝜓)       (𝜑 → ¬ (𝜒𝜃𝜓))
 
Theoreman3andi 1436 Distribution of conjunction over threefold conjunction. (Contributed by Thierry Arnoux, 8-Apr-2019.)
((𝜑 ∧ (𝜓𝜒𝜃)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒) ∧ (𝜑𝜃)))
 
Theoreman33rean 1437 Rearrange a 9-fold conjunction. (Contributed by Thierry Arnoux, 14-Apr-2019.)
(((𝜑𝜓𝜒) ∧ (𝜃𝜏𝜂) ∧ (𝜁𝜎𝜌)) ↔ ((𝜑𝜏𝜌) ∧ ((𝜓𝜃) ∧ (𝜂𝜎) ∧ (𝜒𝜁))))
 
1.2.11  Logical 'nand' (Sheffer stroke)
 
Syntaxwnan 1438 Extend wff definition to include alternative denial ('nand').
wff (𝜑𝜓)
 
Definitiondf-nan 1439 Define incompatibility, or alternative denial ('not-and' or 'nand'). This is also called the Sheffer stroke, represented by a vertical bar, but we use a different symbol to avoid ambiguity with other uses of the vertical bar. In the second edition of Principia Mathematica (1927), Russell and Whitehead used the Sheffer stroke and suggested it as a replacement for the "or" and "not" operations of the first edition. However, in practice, "or" and "not" are more widely used. After we define the constant true (df-tru 1477) and the constant false (df-fal 1480), we will be able to prove these truth table values: ((⊤ ⊼ ⊤) ↔ ⊥) (trunantru 1514), ((⊤ ⊼ ⊥) ↔ ⊤) (trunanfal 1515), ((⊥ ⊼ ⊤) ↔ ⊤) (falnantru 1516), and ((⊥ ⊼ ⊥) ↔ ⊤) (falnanfal 1517). Contrast with (df-an 384), (df-or 383), (wi 4), and (df-xor 1456) . (Contributed by Jeff Hoffman, 19-Nov-2007.)
((𝜑𝜓) ↔ ¬ (𝜑𝜓))
 
Theoremnanan 1440 Write 'and' in terms of 'nand'. (Contributed by Mario Carneiro, 9-May-2015.)
((𝜑𝜓) ↔ ¬ (𝜑𝜓))
 
Theoremnancom 1441 The 'nand' operator commutes. (Contributed by Mario Carneiro, 9-May-2015.) (Proof shortened by Wolf Lammen, 7-Mar-2020.)
((𝜑𝜓) ↔ (𝜓𝜑))
 
Theoremnannan 1442 Lemma for handling nested 'nand's. (Contributed by Jeff Hoffman, 19-Nov-2007.) (Proof shortened by Wolf Lammen, 7-Mar-2020.)
((𝜑 ⊼ (𝜒𝜓)) ↔ (𝜑 → (𝜒𝜓)))
 
Theoremnanim 1443 Show equivalence between implication and the Nicod version. To derive nic-dfim 1584, apply nanbi 1445. (Contributed by Jeff Hoffman, 19-Nov-2007.)
((𝜑𝜓) ↔ (𝜑 ⊼ (𝜓𝜓)))
 
Theoremnannot 1444 Show equivalence between negation and the Nicod version. To derive nic-dfneg 1585, apply nanbi 1445. (Contributed by Jeff Hoffman, 19-Nov-2007.)
𝜓 ↔ (𝜓𝜓))
 
Theoremnanbi 1445 Show equivalence between the biconditional and the Nicod version. (Contributed by Jeff Hoffman, 19-Nov-2007.) (Proof shortened by Wolf Lammen, 27-Jun-2020.)
((𝜑𝜓) ↔ ((𝜑𝜓) ⊼ ((𝜑𝜑) ⊼ (𝜓𝜓))))
 
Theoremnanbi1 1446 Introduce a right anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)
((𝜑𝜓) → ((𝜑𝜒) ↔ (𝜓𝜒)))
 
Theoremnanbi2 1447 Introduce a left anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)
((𝜑𝜓) → ((𝜒𝜑) ↔ (𝜒𝜓)))
 
Theoremnanbi12 1448 Join two logical equivalences with anti-conjunction. (Contributed by SF, 2-Jan-2018.)
(((𝜑𝜓) ∧ (𝜒𝜃)) → ((𝜑𝜒) ↔ (𝜓𝜃)))
 
Theoremnanbi1i 1449 Introduce a right anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)
(𝜑𝜓)       ((𝜑𝜒) ↔ (𝜓𝜒))
 
Theoremnanbi2i 1450 Introduce a left anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)
(𝜑𝜓)       ((𝜒𝜑) ↔ (𝜒𝜓))
 
Theoremnanbi12i 1451 Join two logical equivalences with anti-conjunction. (Contributed by SF, 2-Jan-2018.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) ↔ (𝜓𝜃))
 
Theoremnanbi1d 1452 Introduce a right anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜃)))
 
Theoremnanbi2d 1453 Introduce a left anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜃𝜓) ↔ (𝜃𝜒)))
 
Theoremnanbi12d 1454 Join two logical equivalences with anti-conjunction. (Contributed by Scott Fenton, 2-Jan-2018.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))
 
1.2.12  Logical 'xor'
 
Syntaxwxo 1455 Extend wff definition to include exclusive disjunction ('xor').
wff (𝜑𝜓)
 
Definitiondf-xor 1456 Define exclusive disjunction (logical 'xor'). Return true if either the left or right, but not both, are true. After we define the constant true (df-tru 1477) and the constant false (df-fal 1480), we will be able to prove these truth table values: ((⊤ ⊻ ⊤) ↔ ⊥) (truxortru 1518), ((⊤ ⊻ ⊥) ↔ ⊤) (truxorfal 1519), ((⊥ ⊻ ⊤) ↔ ⊤) (falxortru 1520), and ((⊥ ⊻ ⊥) ↔ ⊥) (falxorfal 1521). Contrast with (df-an 384), (df-or 383), (wi 4), and (df-nan 1439) . (Contributed by FL, 22-Nov-2010.)
((𝜑𝜓) ↔ ¬ (𝜑𝜓))
 
Theoremxnor 1457 Two ways to write XNOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
((𝜑𝜓) ↔ ¬ (𝜑𝜓))
 
Theoremxorcom 1458 The connector is commutative. (Contributed by Mario Carneiro, 4-Sep-2016.)
((𝜑𝜓) ↔ (𝜓𝜑))
 
Theoremxorass 1459 The connector is associative. (Contributed by FL, 22-Nov-2010.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Proof shortened by Wolf Lammen, 20-Jun-2020.)
(((𝜑𝜓) ⊻ 𝜒) ↔ (𝜑 ⊻ (𝜓𝜒)))
 
Theoremexcxor 1460 This tautology shows that xor is really exclusive. (Contributed by FL, 22-Nov-2010.)
((𝜑𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓)))
 
Theoremxor2 1461 Two ways to express "exclusive or." (Contributed by Mario Carneiro, 4-Sep-2016.)
((𝜑𝜓) ↔ ((𝜑𝜓) ∧ ¬ (𝜑𝜓)))
 
Theoremxoror 1462 XOR implies OR. (Contributed by BJ, 19-Apr-2019.)
((𝜑𝜓) → (𝜑𝜓))
 
Theoremxornan 1463 XOR implies NAND. (Contributed by BJ, 19-Apr-2019.)
((𝜑𝜓) → ¬ (𝜑𝜓))
 
Theoremxornan2 1464 XOR implies NAND (written with the connector). (Contributed by BJ, 19-Apr-2019.)
((𝜑𝜓) → (𝜑𝜓))
 
Theoremxorneg2 1465 The connector is negated under negation of one argument. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 27-Jun-2020.)
((𝜑 ⊻ ¬ 𝜓) ↔ ¬ (𝜑𝜓))
 
Theoremxorneg1 1466 The connector is negated under negation of one argument. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 27-Jun-2020.)
((¬ 𝜑𝜓) ↔ ¬ (𝜑𝜓))
 
Theoremxorneg 1467 The connector is unchanged under negation of both arguments. (Contributed by Mario Carneiro, 4-Sep-2016.)
((¬ 𝜑 ⊻ ¬ 𝜓) ↔ (𝜑𝜓))
 
Theoremxorbi12i 1468 Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) ↔ (𝜓𝜃))
 
Theoremxorbi12d 1469 Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))
 
Theoremanxordi 1470 Conjunction distributes over exclusive-or. In intuitionistic logic this assertion is also true, even though xordi 934 does not necessarily hold, in part because the usual definition of xor is subtly different in intuitionistic logic. (Contributed by David A. Wheeler, 7-Oct-2018.)
((𝜑 ∧ (𝜓𝜒)) ↔ ((𝜑𝜓) ⊻ (𝜑𝜒)))
 
Theoremxorexmid 1471 Exclusive-or variant of the law of the excluded middle (exmid 429). This statement is ancient, going back to at least Stoic logic. This statement does not necessarily hold in intuitionistic logic. (Contributed by David A. Wheeler, 23-Feb-2019.)
(𝜑 ⊻ ¬ 𝜑)
 
1.2.13  True and false constants
 
1.2.13.1  Universal quantifier for use by df-tru

Even though it isn't ordinarily part of propositional calculus, the universal quantifier is introduced here so that the soundness of definition df-tru 1477 can be checked by the same algorithm that is used for predicate calculus. Its first real use is in definition df-ex 1695 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 1482 may be adopted and this subsection moved down to the start of the subsection with wex 1694 below. However, the use of dftru2 1482 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid.

 
Syntaxwal 1472 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
 
1.2.13.2  Equality predicate for use by df-tru

Even though it isn't ordinarily part of propositional calculus, the equality predicate = is introduced here so that the soundness of definition df-tru 1477 can be checked by the same algorithm as is used for predicate calculus. Its first real use is in theorem equs3 1825 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 1482 may be adopted and this subsection moved down to just above weq 1824 below. However, the use of dftru2 1482 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid.

 
Syntaxcv 1473 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 {𝑦𝑦𝑥} is a class by cab 2500. Since (when 𝑦 is distinct from 𝑥) we have 𝑥 = {𝑦𝑦𝑥} by cvjust 2509, we can argue that the syntax "class 𝑥 " can be viewed as an abbreviation for "class {𝑦𝑦𝑥}". 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 1473 as a "type conversion" from a setvar variable to a class variable, keep in mind that cv 1473 is intrinsically no different from any other class-building syntax such as cab 2500, cun 3442, or c0 3777.

For a general discussion of the theory of classes and the role of cv 1473, see 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 1824 of predicate calculus from the wceq 1474 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
 
Syntaxwceq 1474 Extend wff definition to include class equality.

For a general discussion of the theory of classes, see 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 1824 of predicate calculus in terms of the wceq 1474 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 1824 or wceq 1474, 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 2507 for more information on the set theory usage of wceq 1474.)

wff 𝐴 = 𝐵
 
1.2.13.3  Define the true and false constants
 
Syntaxwtru 1475 The constant is a wff.
wff
 
Theoremtrujust 1476 Soundness justification theorem for df-tru 1477. (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))
 
Definitiondf-tru 1477 Definition of the truth value "true", or "verum", denoted by . This is a tautology, as proved by tru 1478. In this definition, an instance of id 22 is used as the definiens, although any tautology, such as an axiom, can be used in its place. This particular id 22 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 1478, and other proofs should depend on tru 1478 (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))
 
Theoremtru 1478 The truth value is provable. (Contributed by Anthony Hart, 13-Oct-2010.)
 
Syntaxwfal 1479 The constant is a wff.
wff
 
Definitiondf-fal 1480 Definition of the truth value "false", or "falsum", denoted by . See also df-tru 1477. (Contributed by Anthony Hart, 22-Oct-2010.)
(⊥ ↔ ¬ ⊤)
 
Theoremfal 1481 The truth value is refutable. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Mel L. O'Cat, 11-Mar-2012.)
¬ ⊥
 
Theoremdftru2 1482 An alternate definition of "true". (Contributed by Anthony Hart, 13-Oct-2010.) (Revised by BJ, 12-Jul-2019.) (New usage is discouraged.)
(⊤ ↔ (𝜑𝜑))
 
Theoremtrud 1483 Eliminate as an antecedent. A proposition implied by is true. (Contributed by Mario Carneiro, 13-Mar-2014.)
(⊤ → 𝜑)       𝜑
 
Theoremtbtru 1484 A proposition is equivalent to itself being equivalent to . (Contributed by Anthony Hart, 14-Aug-2011.)
(𝜑 ↔ (𝜑 ↔ ⊤))
 
Theoremnbfal 1485 The negation of a proposition is equivalent to itself being equivalent to . (Contributed by Anthony Hart, 14-Aug-2011.)
𝜑 ↔ (𝜑 ↔ ⊥))
 
Theorembitru 1486 A theorem is equivalent to truth. (Contributed by Mario Carneiro, 9-May-2015.)
𝜑       (𝜑 ↔ ⊤)
 
Theorembifal 1487 A contradiction is equivalent to falsehood. (Contributed by Mario Carneiro, 9-May-2015.)
¬ 𝜑       (𝜑 ↔ ⊥)
 
Theoremfalim 1488 The truth value implies anything. Also called the "principle of explosion", or "ex falso [sequitur]] quodlibet" (Latin for "from falsehood, anything [follows]]"). (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.)
(⊥ → 𝜑)
 
Theoremfalimd 1489 The truth value implies anything. (Contributed by Mario Carneiro, 9-Feb-2017.)
((𝜑 ∧ ⊥) → 𝜓)
 
Theorema1tru 1490 Anything implies . (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.)
(𝜑 → ⊤)
 
Theoremtruan 1491 True can be removed from a conjunction. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Wolf Lammen, 21-Jul-2019.)
((⊤ ∧ 𝜑) ↔ 𝜑)
 
Theoremdfnot 1492 Given falsum , we can define the negation of a wff 𝜑 as the statement that follows from assuming 𝜑. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof shortened by Wolf Lammen, 21-Jul-2019.)
𝜑 ↔ (𝜑 → ⊥))
 
Theoreminegd 1493 Negation introduction rule from natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)
((𝜑𝜓) → ⊥)       (𝜑 → ¬ 𝜓)
 
Theoremefald 1494 Deduction based on reductio ad absurdum. (Contributed by Mario Carneiro, 9-Feb-2017.)
((𝜑 ∧ ¬ 𝜓) → ⊥)       (𝜑𝜓)
 
Theorempm2.21fal 1495 If a wff and its negation are provable, then falsum is provable. (Contributed by Mario Carneiro, 9-Feb-2017.)
(𝜑𝜓)    &   (𝜑 → ¬ 𝜓)       (𝜑 → ⊥)
 
1.2.14  Truth tables

Some sources define operations on true/false values using truth tables. These tables show the results of their operations for all possible combinations of true () and false (). Here we show that our definitions and axioms produce equivalent results for (conjunction aka logical 'and') df-an 384, (disjunction aka logical inclusive 'or') df-or 383, (implies) wi 4, ¬ (not) wn 3, (logical equivalence) df-bi 195, (nand aka Sheffer stroke) df-nan 1439, and (exclusive or) df-xor 1456.

 
Theoremtruantru 1496 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊤ ∧ ⊤) ↔ ⊤)
 
Theoremtruanfal 1497 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊤ ∧ ⊥) ↔ ⊥)
 
Theoremfalantru 1498 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊥ ∧ ⊤) ↔ ⊥)
 
Theoremfalanfal 1499 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
((⊥ ∧ ⊥) ↔ ⊥)
 
Theoremtruortru 1500 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((⊤ ∨ ⊤) ↔ ⊤)
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