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Theorem List for Metamath Proof Explorer - 1401-1500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem3anbi2d 1401 Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜃𝜓𝜏) ↔ (𝜃𝜒𝜏)))

Theorem3anbi3d 1402 Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜃𝜏𝜓) ↔ (𝜃𝜏𝜒)))

Theorem3anim123d 1403 Deduction joining 3 implications to form implication of conjunctions. (Contributed by NM, 24-Feb-2005.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))    &   (𝜑 → (𝜂𝜁))       (𝜑 → ((𝜓𝜃𝜂) → (𝜒𝜏𝜁)))

Theorem3orim123d 1404 Deduction joining 3 implications to form implication of disjunctions. (Contributed by NM, 4-Apr-1997.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))    &   (𝜑 → (𝜂𝜁))       (𝜑 → ((𝜓𝜃𝜂) → (𝜒𝜏𝜁)))

Theoreman6 1405 Rearrangement of 6 conjuncts. (Contributed by NM, 13-Mar-1995.)
(((𝜑𝜓𝜒) ∧ (𝜃𝜏𝜂)) ↔ ((𝜑𝜃) ∧ (𝜓𝜏) ∧ (𝜒𝜂)))

Theorem3an6 1406 Analogue of an4 864 for triple conjunction. (Contributed by Scott Fenton, 16-Mar-2011.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂)) ↔ ((𝜑𝜒𝜏) ∧ (𝜓𝜃𝜂)))

Theorem3or6 1407 Analogue of or4 550 for triple conjunction. (Contributed by Scott Fenton, 16-Mar-2011.)
(((𝜑𝜓) ∨ (𝜒𝜃) ∨ (𝜏𝜂)) ↔ ((𝜑𝜒𝜏) ∨ (𝜓𝜃𝜂)))

Theoremmp3an1 1408 An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.)
𝜑    &   ((𝜑𝜓𝜒) → 𝜃)       ((𝜓𝜒) → 𝜃)

Theoremmp3an2 1409 An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.)
𝜓    &   ((𝜑𝜓𝜒) → 𝜃)       ((𝜑𝜒) → 𝜃)

Theoremmp3an3 1410 An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.)
𝜒    &   ((𝜑𝜓𝜒) → 𝜃)       ((𝜑𝜓) → 𝜃)

Theoremmp3an12 1411 An inference based on modus ponens. (Contributed by NM, 13-Jul-2005.)
𝜑    &   𝜓    &   ((𝜑𝜓𝜒) → 𝜃)       (𝜒𝜃)

Theoremmp3an13 1412 An inference based on modus ponens. (Contributed by NM, 14-Jul-2005.)
𝜑    &   𝜒    &   ((𝜑𝜓𝜒) → 𝜃)       (𝜓𝜃)

Theoremmp3an23 1413 An inference based on modus ponens. (Contributed by NM, 14-Jul-2005.)
𝜓    &   𝜒    &   ((𝜑𝜓𝜒) → 𝜃)       (𝜑𝜃)

Theoremmp3an1i 1414 An inference based on modus ponens. (Contributed by NM, 5-Jul-2005.)
𝜓    &   (𝜑 → ((𝜓𝜒𝜃) → 𝜏))       (𝜑 → ((𝜒𝜃) → 𝜏))

Theoremmp3anl1 1415 An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.)
𝜑    &   (((𝜑𝜓𝜒) ∧ 𝜃) → 𝜏)       (((𝜓𝜒) ∧ 𝜃) → 𝜏)

Theoremmp3anl2 1416 An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.)
𝜓    &   (((𝜑𝜓𝜒) ∧ 𝜃) → 𝜏)       (((𝜑𝜒) ∧ 𝜃) → 𝜏)

Theoremmp3anl3 1417 An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.)
𝜒    &   (((𝜑𝜓𝜒) ∧ 𝜃) → 𝜏)       (((𝜑𝜓) ∧ 𝜃) → 𝜏)

Theoremmp3anr1 1418 An inference based on modus ponens. (Contributed by NM, 4-Nov-2006.)
𝜓    &   ((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜏)       ((𝜑 ∧ (𝜒𝜃)) → 𝜏)

Theoremmp3anr2 1419 An inference based on modus ponens. (Contributed by NM, 24-Nov-2006.)
𝜒    &   ((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜏)       ((𝜑 ∧ (𝜓𝜃)) → 𝜏)

Theoremmp3anr3 1420 An inference based on modus ponens. (Contributed by NM, 19-Oct-2007.)
𝜃    &   ((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜏)       ((𝜑 ∧ (𝜓𝜒)) → 𝜏)

Theoremmp3an 1421 An inference based on modus ponens. (Contributed by NM, 14-May-1999.)
𝜑    &   𝜓    &   𝜒    &   ((𝜑𝜓𝜒) → 𝜃)       𝜃

Theoremmpd3an3 1422 An inference based on modus ponens. (Contributed by NM, 8-Nov-2007.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜓𝜒) → 𝜃)       ((𝜑𝜓) → 𝜃)

Theoremmpd3an23 1423 An inference based on modus ponens. (Contributed by NM, 4-Dec-2006.)
(𝜑𝜓)    &   (𝜑𝜒)    &   ((𝜑𝜓𝜒) → 𝜃)       (𝜑𝜃)

Theoremmp3and 1424 A deduction based on modus ponens. (Contributed by Mario Carneiro, 24-Dec-2016.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑 → ((𝜓𝜒𝜃) → 𝜏))       (𝜑𝜏)

Theoremmp3an12i 1425 mp3an 1421 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 1426 mp3an 1421 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 1427 mp3an 1421 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 1428 An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.)
𝜑    &   (𝜓𝜒)    &   ((𝜓𝜃) → 𝜏)    &   ((𝜑𝜒𝜏) → 𝜂)       ((𝜓𝜃) → 𝜂)

Theorembiimp3a 1429 Infer implication from a logical equivalence. Similar to biimpa 501. (Contributed by NM, 4-Sep-2005.)
((𝜑𝜓) → (𝜒𝜃))       ((𝜑𝜓𝜒) → 𝜃)

Theorembiimp3ar 1430 Infer implication from a logical equivalence. Similar to biimpar 502. (Contributed by NM, 2-Jan-2009.)
((𝜑𝜓) → (𝜒𝜃))       ((𝜑𝜓𝜃) → 𝜒)

Theorem3anandis 1431 Inference that undistributes a triple conjunction in the antecedent. (Contributed by NM, 18-Apr-2007.)
(((𝜑𝜓) ∧ (𝜑𝜒) ∧ (𝜑𝜃)) → 𝜏)       ((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜏)

Theorem3anandirs 1432 Inference that undistributes a triple conjunction in the antecedent. (Contributed by NM, 25-Jul-2006.)
(((𝜑𝜃) ∧ (𝜓𝜃) ∧ (𝜒𝜃)) → 𝜏)       (((𝜑𝜓𝜒) ∧ 𝜃) → 𝜏)

Theoremecase23d 1433 Deduction for elimination by cases. (Contributed by NM, 22-Apr-1994.)
(𝜑 → ¬ 𝜒)    &   (𝜑 → ¬ 𝜃)    &   (𝜑 → (𝜓𝜒𝜃))       (𝜑𝜓)

Theorem3ecase 1434 Inference for elimination by cases. (Contributed by NM, 13-Jul-2005.)
𝜑𝜃)    &   𝜓𝜃)    &   𝜒𝜃)    &   ((𝜑𝜓𝜒) → 𝜃)       𝜃

Theorem3bior1fd 1435 A disjunction is equivalent to a threefold disjunction with single falsehood, analogous to biorf 420. (Contributed by Alexander van der Vekens, 8-Sep-2017.)
(𝜑 → ¬ 𝜃)       (𝜑 → ((𝜒𝜓) ↔ (𝜃𝜒𝜓)))

Theorem3bior1fand 1436 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 1437 A wff is equivalent to its threefold disjunction with double falsehood, analogous to biorf 420. (Contributed by Alexander van der Vekens, 8-Sep-2017.)
(𝜑 → ¬ 𝜃)    &   (𝜑 → ¬ 𝜒)       (𝜑 → (𝜓 ↔ (𝜃𝜒𝜓)))

Theorem3biant1d 1438 A conjunction is equivalent to a threefold conjunction with single truth, analogous to biantrud 528. (Contributed by Alexander van der Vekens, 26-Sep-2017.)
(𝜑𝜃)       (𝜑 → ((𝜒𝜓) ↔ (𝜃𝜒𝜓)))

Theoremintn3an1d 1439 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 1440 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 1441 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 1442 Distribution of conjunction over threefold conjunction. (Contributed by Thierry Arnoux, 8-Apr-2019.)
((𝜑 ∧ (𝜓𝜒𝜃)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒) ∧ (𝜑𝜃)))

Theoreman33rean 1443 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 1444 Extend wff definition to include alternative denial ('nand').
wff (𝜑𝜓)

Definitiondf-nan 1445 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 1483) and the constant false (df-fal 1486), we will be able to prove these truth table values: ((⊤ ⊼ ⊤) ↔ ⊥) (trunantru 1521), ((⊤ ⊼ ⊥) ↔ ⊤) (trunanfal 1522), ((⊥ ⊼ ⊤) ↔ ⊤) (falnantru 1523), and ((⊥ ⊼ ⊥) ↔ ⊤) (falnanfal 1524). Contrast with (df-an 386), (df-or 385), (wi 4), and (df-xor 1462) . (Contributed by Jeff Hoffman, 19-Nov-2007.)
((𝜑𝜓) ↔ ¬ (𝜑𝜓))

Theoremnanan 1446 Write 'and' in terms of 'nand'. (Contributed by Mario Carneiro, 9-May-2015.)
((𝜑𝜓) ↔ ¬ (𝜑𝜓))

Theoremnancom 1447 The 'nand' operator commutes. (Contributed by Mario Carneiro, 9-May-2015.) (Proof shortened by Wolf Lammen, 7-Mar-2020.)
((𝜑𝜓) ↔ (𝜓𝜑))

Theoremnannan 1448 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 1449 Show equivalence between implication and the Nicod version. To derive nic-dfim 1591, apply nanbi 1451. (Contributed by Jeff Hoffman, 19-Nov-2007.)
((𝜑𝜓) ↔ (𝜑 ⊼ (𝜓𝜓)))

Theoremnannot 1450 Show equivalence between negation and the Nicod version. To derive nic-dfneg 1592, apply nanbi 1451. (Contributed by Jeff Hoffman, 19-Nov-2007.)
𝜓 ↔ (𝜓𝜓))

Theoremnanbi 1451 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 1452 Introduce a right anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)
((𝜑𝜓) → ((𝜑𝜒) ↔ (𝜓𝜒)))

Theoremnanbi2 1453 Introduce a left anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)
((𝜑𝜓) → ((𝜒𝜑) ↔ (𝜒𝜓)))

Theoremnanbi12 1454 Join two logical equivalences with anti-conjunction. (Contributed by SF, 2-Jan-2018.)
(((𝜑𝜓) ∧ (𝜒𝜃)) → ((𝜑𝜒) ↔ (𝜓𝜃)))

Theoremnanbi1i 1455 Introduce a right anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)
(𝜑𝜓)       ((𝜑𝜒) ↔ (𝜓𝜒))

Theoremnanbi2i 1456 Introduce a left anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)
(𝜑𝜓)       ((𝜒𝜑) ↔ (𝜒𝜓))

Theoremnanbi12i 1457 Join two logical equivalences with anti-conjunction. (Contributed by SF, 2-Jan-2018.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) ↔ (𝜓𝜃))

Theoremnanbi1d 1458 Introduce a right anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜃)))

Theoremnanbi2d 1459 Introduce a left anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜃𝜓) ↔ (𝜃𝜒)))

Theoremnanbi12d 1460 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 1461 Extend wff definition to include exclusive disjunction ('xor').
wff (𝜑𝜓)

Definitiondf-xor 1462 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 1483) and the constant false (df-fal 1486), we will be able to prove these truth table values: ((⊤ ⊻ ⊤) ↔ ⊥) (truxortru 1525), ((⊤ ⊻ ⊥) ↔ ⊤) (truxorfal 1526), ((⊥ ⊻ ⊤) ↔ ⊤) (falxortru 1527), and ((⊥ ⊻ ⊥) ↔ ⊥) (falxorfal 1528). Contrast with (df-an 386), (df-or 385), (wi 4), and (df-nan 1445) . (Contributed by FL, 22-Nov-2010.)
((𝜑𝜓) ↔ ¬ (𝜑𝜓))

Theoremxnor 1463 Two ways to write XNOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
((𝜑𝜓) ↔ ¬ (𝜑𝜓))

Theoremxorcom 1464 The connector is commutative. (Contributed by Mario Carneiro, 4-Sep-2016.)
((𝜑𝜓) ↔ (𝜓𝜑))

Theoremxorass 1465 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 1466 This tautology shows that xor is really exclusive. (Contributed by FL, 22-Nov-2010.)
((𝜑𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓)))

Theoremxor2 1467 Two ways to express "exclusive or." (Contributed by Mario Carneiro, 4-Sep-2016.)
((𝜑𝜓) ↔ ((𝜑𝜓) ∧ ¬ (𝜑𝜓)))

Theoremxoror 1468 XOR implies OR. (Contributed by BJ, 19-Apr-2019.)
((𝜑𝜓) → (𝜑𝜓))

Theoremxornan 1469 XOR implies NAND. (Contributed by BJ, 19-Apr-2019.)
((𝜑𝜓) → ¬ (𝜑𝜓))

Theoremxornan2 1470 XOR implies NAND (written with the connector). (Contributed by BJ, 19-Apr-2019.)
((𝜑𝜓) → (𝜑𝜓))

Theoremxorneg2 1471 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 1472 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 1473 The connector is unchanged under negation of both arguments. (Contributed by Mario Carneiro, 4-Sep-2016.)
((¬ 𝜑 ⊻ ¬ 𝜓) ↔ (𝜑𝜓))

Theoremxorbi12i 1474 Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) ↔ (𝜓𝜃))

Theoremxorbi12d 1475 Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))

Theoremanxordi 1476 Conjunction distributes over exclusive-or. In intuitionistic logic this assertion is also true, even though xordi 936 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 1477 Exclusive-or variant of the law of the excluded middle (exmid 431). 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 1483 can be checked by the same algorithm that is used for predicate calculus. Its first real use is in definition df-ex 1702 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 1488 may be adopted and this subsection moved down to the start of the subsection with wex 1701 below. However, the use of dftru2 1488 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid.

Syntaxwal 1478 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 1483 can be checked by the same algorithm as is used for predicate calculus. Its first real use is in theorem equs3 1872 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 1488 may be adopted and this subsection moved down to just above weq 1871 below. However, the use of dftru2 1488 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid.

Syntaxcv 1479 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 2607. Since (when 𝑦 is distinct from 𝑥) we have 𝑥 = {𝑦𝑦𝑥} by cvjust 2616, 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 1479 as a "type conversion" from a setvar variable to a class variable, keep in mind that cv 1479 is intrinsically no different from any other class-building syntax such as cab 2607, cun 3558, or c0 3897.

For a general discussion of the theory of classes and the role of cv 1479, 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 1871 of predicate calculus from the wceq 1480 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 1480 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 1871 of predicate calculus in terms of the wceq 1480 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 1871 or wceq 1480, 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 2614 for more information on the set theory usage of wceq 1480.)

wff 𝐴 = 𝐵

1.2.13.3  Define the true and false constants

Syntaxwtru 1481 The constant is a wff.
wff

Theoremtrujust 1482 Soundness justification theorem for df-tru 1483. (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 1483 Definition of the truth value "true", or "verum", denoted by . This is a tautology, as proved by tru 1484. 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 1484, and other proofs should depend on tru 1484 (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.) Use tru 1484 instead. (New usage is discouraged.)
(⊤ ↔ (∀𝑥 𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥))

Theoremtru 1484 The truth value is provable. (Contributed by Anthony Hart, 13-Oct-2010.)

Syntaxwfal 1485 The constant is a wff.
wff

Definitiondf-fal 1486 Definition of the truth value "false", or "falsum", denoted by . See also df-tru 1483. (Contributed by Anthony Hart, 22-Oct-2010.)
(⊥ ↔ ¬ ⊤)

Theoremfal 1487 The truth value is refutable. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Mel L. O'Cat, 11-Mar-2012.)
¬ ⊥

Theoremdftru2 1488 An alternate definition of "true". (Contributed by Anthony Hart, 13-Oct-2010.) (Revised by BJ, 12-Jul-2019.) (New usage is discouraged.)
(⊤ ↔ (𝜑𝜑))

Theoremtrut 1489 A proposition is equivalent to it being implied by . Closed form of trud 1490. Dual of dfnot 1499. It is to tbtru 1491 what a1bi 352 is to tbt 359. (Contributed by BJ, 26-Oct-2019.)
(𝜑 ↔ (⊤ → 𝜑))

Theoremtrud 1490 Eliminate as an antecedent. A proposition implied by is true. (Contributed by Mario Carneiro, 13-Mar-2014.)
(⊤ → 𝜑)       𝜑

Theoremtbtru 1491 A proposition is equivalent to itself being equivalent to . (Contributed by Anthony Hart, 14-Aug-2011.)
(𝜑 ↔ (𝜑 ↔ ⊤))

Theoremnbfal 1492 The negation of a proposition is equivalent to itself being equivalent to . (Contributed by Anthony Hart, 14-Aug-2011.)
𝜑 ↔ (𝜑 ↔ ⊥))

Theorembitru 1493 A theorem is equivalent to truth. (Contributed by Mario Carneiro, 9-May-2015.)
𝜑       (𝜑 ↔ ⊤)

Theorembifal 1494 A contradiction is equivalent to falsehood. (Contributed by Mario Carneiro, 9-May-2015.)
¬ 𝜑       (𝜑 ↔ ⊥)

Theoremfalim 1495 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 1496 The truth value implies anything. (Contributed by Mario Carneiro, 9-Feb-2017.)
((𝜑 ∧ ⊥) → 𝜓)

Theorema1tru 1497 Anything implies . (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.)
(𝜑 → ⊤)

Theoremtruan 1498 True can be removed from a conjunction. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Wolf Lammen, 21-Jul-2019.)
((⊤ ∧ 𝜑) ↔ 𝜑)

Theoremdfnot 1499 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 1500 Negation introduction rule from natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)
((𝜑𝜓) → ⊥)       (𝜑 → ¬ 𝜓)

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