Home Intuitionistic Logic ExplorerTheorem List (p. 16 of 102) < Previous  Next > Browser slow? Try the Unicode version. Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 1501-1600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremalrimd 1501 Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)

Theoremeximdh 1502 Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 20-May-1996.)

Theoremeximd 1503 Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)

Theoremnexd 1504 Deduction for generalization rule for negated wff. (Contributed by NM, 2-Jan-2002.)

Theoremexbidh 1505 Formula-building rule for existential quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.)

Theoremalbid 1506 Formula-building rule for universal quantifier (deduction rule). (Contributed by Mario Carneiro, 24-Sep-2016.)

Theoremexbid 1507 Formula-building rule for existential quantifier (deduction rule). (Contributed by Mario Carneiro, 24-Sep-2016.)

Theoremexsimpl 1508 Simplification of an existentially quantified conjunction. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremexsimpr 1509 Simplification of an existentially quantified conjunction. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremalexdc 1510 Theorem 19.6 of [Margaris] p. 89, given a decidability condition. The forward direction holds for all propositions, as seen at alexim 1536. (Contributed by Jim Kingdon, 2-Jun-2018.)
DECID

Theorem19.29 1511 Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Theorem19.29r 1512 Variation of Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.)

Theorem19.29r2 1513 Variation of Theorem 19.29 of [Margaris] p. 90 with double quantification. (Contributed by NM, 3-Feb-2005.)

Theorem19.29x 1514 Variation of Theorem 19.29 of [Margaris] p. 90 with mixed quantification. (Contributed by NM, 11-Feb-2005.)

Theorem19.35-1 1515 Forward direction of Theorem 19.35 of [Margaris] p. 90. The converse holds for classical logic but not (for all propositions) in intuitionistic logic (Contributed by Mario Carneiro, 2-Feb-2015.)

Theorem19.35i 1516 Inference from Theorem 19.35 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)

Theorem19.25 1517 Theorem 19.25 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)

Theorem19.30dc 1518 Theorem 19.30 of [Margaris] p. 90, with an additional decidability condition. (Contributed by Jim Kingdon, 21-Jul-2018.)
DECID

Theorem19.43 1519 Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 2-Feb-2015.)

Theorem19.33b2 1520 The antecedent provides a condition implying the converse of 19.33 1373. Compare Theorem 19.33 of [Margaris] p. 90. This variation of 19.33bdc 1521 is intuitionistically valid without a decidability condition. (Contributed by Mario Carneiro, 2-Feb-2015.)

Theorem19.33bdc 1521 Converse of 19.33 1373 given and a decidability condition. Compare Theorem 19.33 of [Margaris] p. 90. For a version which does not require a decidability condition, see 19.33b2 1520 (Contributed by Jim Kingdon, 23-Apr-2018.)
DECID

Theorem19.40 1522 Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.40-2 1523 Theorem *11.42 in [WhiteheadRussell] p. 163. Theorem 19.40 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)

Theoremexintrbi 1524 Add/remove a conjunct in the scope of an existential quantifier. (Contributed by Raph Levien, 3-Jul-2006.)

Theoremexintr 1525 Introduce a conjunct in the scope of an existential quantifier. (Contributed by NM, 11-Aug-1993.)

Theoremalsyl 1526 Theorem *10.3 in [WhiteheadRussell] p. 150. (Contributed by Andrew Salmon, 8-Jun-2011.)

Theoremhbex 1527 If is not free in , it is not free in . (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)

Theoremnfex 1528 If is not free in , it is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.)

Theorem19.2 1529 Theorem 19.2 of [Margaris] p. 89, generalized to use two setvar variables. (Contributed by O'Cat, 31-Mar-2008.)

Theoremi19.24 1530 Theorem 19.24 of [Margaris] p. 90, with an additional hypothesis. The hypothesis is the converse of 19.35-1 1515, and is a theorem of classical logic, but in intuitionistic logic it will only be provable for some propositions. (Contributed by Jim Kingdon, 22-Jul-2018.)

Theoremi19.39 1531 Theorem 19.39 of [Margaris] p. 90, with an additional hypothesis. The hypothesis is the converse of 19.35-1 1515, and is a theorem of classical logic, but in intuitionistic logic it will only be provable for some propositions. (Contributed by Jim Kingdon, 22-Jul-2018.)

Theorem19.9ht 1532 A closed version of one direction of 19.9 1535. (Contributed by NM, 5-Aug-1993.)

Theorem19.9t 1533 A closed version of 19.9 1535. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortended by Wolf Lammen, 30-Dec-2017.)

Theorem19.9h 1534 A wff may be existentially quantified with a variable not free in it. Theorem 19.9 of [Margaris] p. 89. (Contributed by FL, 24-Mar-2007.)

Theorem19.9 1535 A wff may be existentially quantified with a variable not free in it. Theorem 19.9 of [Margaris] p. 89. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.)

Theoremalexim 1536 One direction of theorem 19.6 of [Margaris] p. 89. The converse holds given a decidability condition, as seen at alexdc 1510. (Contributed by Jim Kingdon, 2-Jul-2018.)

Theoremexnalim 1537 One direction of Theorem 19.14 of [Margaris] p. 90. In classical logic the converse also holds. (Contributed by Jim Kingdon, 15-Jul-2018.)

Theoremexanaliim 1538 A transformation of quantifiers and logical connectives. In classical logic the converse also holds. (Contributed by Jim Kingdon, 15-Jul-2018.)

Theoremalexnim 1539 A relationship between two quantifiers and negation. (Contributed by Jim Kingdon, 27-Aug-2018.)

Theoremax6blem 1540 If is not free in , it is not free in . This theorem doesn't use ax6b 1541 compared to hbnt 1543. (Contributed by GD, 27-Jan-2018.)

Theoremax6b 1541 Quantified Negation. Axiom C5-2 of [Monk2] p. 113.

(Contributed by GD, 27-Jan-2018.)

Theoremhbn1 1542 is not free in . (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 18-Aug-2014.)

Theoremhbnt 1543 Closed theorem version of bound-variable hypothesis builder hbn 1544. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)

Theoremhbn 1544 If is not free in , it is not free in . (Contributed by NM, 5-Aug-1993.)

Theoremhbnd 1545 Deduction form of bound-variable hypothesis builder hbn 1544. (Contributed by NM, 3-Jan-2002.)

Theoremnfnt 1546 If is not free in , then it is not free in . (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 28-Dec-2017.) (Revised by BJ, 24-Jul-2019.)

Theoremnfnd 1547 Deduction associated with nfnt 1546. (Contributed by Mario Carneiro, 24-Sep-2016.)

Theoremnfn 1548 Inference associated with nfnt 1546. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfdc 1549 If is not free in , it is not free in DECID . (Contributed by Jim Kingdon, 11-Mar-2018.)
DECID

Theoremmodal-5 1550 The analog in our predicate calculus of axiom 5 of modal logic S5. (Contributed by NM, 5-Oct-2005.)

Theorem19.9d 1551 A deduction version of one direction of 19.9 1535. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)

Theorem19.9hd 1552 A deduction version of one direction of 19.9 1535. This is an older variation of this theorem; new proofs should use 19.9d 1551. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Theoremexcomim 1553 One direction of Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)

Theoremexcom 1554 Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)

Theorem19.12 1555 Theorem 19.12 of [Margaris] p. 89. Assuming the converse is a mistake sometimes made by beginners! (Contributed by NM, 5-Aug-1993.)

Theorem19.19 1556 Theorem 19.19 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)

Theorem19.21-2 1557 Theorem 19.21 of [Margaris] p. 90 but with 2 quantifiers. (Contributed by NM, 4-Feb-2005.)

Theoremnf2 1558 An alternative definition of df-nf 1350, which does not involve nested quantifiers on the same variable. (Contributed by Mario Carneiro, 24-Sep-2016.)

Theoremnf3 1559 An alternative definition of df-nf 1350. (Contributed by Mario Carneiro, 24-Sep-2016.)

Theoremnf4dc 1560 Variable is effectively not free in iff is always true or always false, given a decidability condition. The reverse direction, nf4r 1561, holds for all propositions. (Contributed by Jim Kingdon, 21-Jul-2018.)
DECID

Theoremnf4r 1561 If is always true or always false, then variable is effectively not free in . The converse holds given a decidability condition, as seen at nf4dc 1560. (Contributed by Jim Kingdon, 21-Jul-2018.)

Theorem19.36i 1562 Inference from Theorem 19.36 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)

Theorem19.36-1 1563 Closed form of 19.36i 1562. One direction of Theorem 19.36 of [Margaris] p. 90. The converse holds in classical logic, but does not hold (for all propositions) in intuitionistic logic. (Contributed by Jim Kingdon, 20-Jun-2018.)

Theorem19.37-1 1564 One direction of Theorem 19.37 of [Margaris] p. 90. The converse holds in classical logic but not, in general, here. (Contributed by Jim Kingdon, 21-Jun-2018.)

Theorem19.37aiv 1565* Inference from Theorem 19.37 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.38 1566 Theorem 19.38 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.23t 1567 Closed form of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)

Theorem19.23 1568 Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)

Theorem19.32dc 1569 Theorem 19.32 of [Margaris] p. 90, where is decidable. (Contributed by Jim Kingdon, 4-Jun-2018.)
DECID

Theorem19.32r 1570 One direction of Theorem 19.32 of [Margaris] p. 90. The converse holds if is decidable, as seen at 19.32dc 1569. (Contributed by Jim Kingdon, 28-Jul-2018.)

Theorem19.31r 1571 One direction of Theorem 19.31 of [Margaris] p. 90. The converse holds in classical logic, but not intuitionistic logic. (Contributed by Jim Kingdon, 28-Jul-2018.)

Theorem19.44 1572 Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)

Theorem19.45 1573 Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)

Theorem19.34 1574 Theorem 19.34 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.41h 1575 Theorem 19.41 of [Margaris] p. 90. New proofs should use 19.41 1576 instead. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.)

Theorem19.41 1576 Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 12-Jan-2018.)

Theorem19.42h 1577 Theorem 19.42 of [Margaris] p. 90. New proofs should use 19.42 1578 instead. (Contributed by NM, 18-Aug-1993.) (New usage is discouraged.)

Theorem19.42 1578 Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.)

Theoremexcom13 1579 Swap 1st and 3rd existential quantifiers. (Contributed by NM, 9-Mar-1995.)

Theoremexrot3 1580 Rotate existential quantifiers. (Contributed by NM, 17-Mar-1995.)

Theoremexrot4 1581 Rotate existential quantifiers twice. (Contributed by NM, 9-Mar-1995.)

Theoremnexr 1582 Inference from 19.8a 1482. (Contributed by Jeff Hankins, 26-Jul-2009.)

Theoremexan 1583 Place a conjunct in the scope of an existential quantifier. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremhbexd 1584 Deduction form of bound-variable hypothesis builder hbex 1527. (Contributed by NM, 2-Jan-2002.)

Theoremeeor 1585 Rearrange existential quantifiers. (Contributed by NM, 8-Aug-1994.)

1.3.8  Equality theorems without distinct variables

Theorema9e 1586 At least one individual exists. This is not a theorem of free logic, which is sound in empty domains. For such a logic, we would add this theorem as an axiom of set theory (Axiom 0 of [Kunen] p. 10). In the system consisting of ax-5 1336 through ax-14 1405 and ax-17 1419, all axioms other than ax-9 1424 are believed to be theorems of free logic, although the system without ax-9 1424 is probably not complete in free logic. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)

Theorema9ev 1587* At least one individual exists. Weaker version of a9e 1586. (Contributed by NM, 3-Aug-2017.)

Theoremax9o 1588 An implication related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)

Theoremequid 1589 Identity law for equality (reflexivity). Lemma 6 of [Tarski] p. 68. This is often an axiom of equality in textbook systems, but we don't need it as an axiom since it can be proved from our other axioms.

This proof is similar to Tarski's and makes use of a dummy variable . It also works in intuitionistic logic, unlike some other possible ways of proving this theorem. (Contributed by NM, 1-Apr-2005.)

Theoremnfequid 1590 Bound-variable hypothesis builder for . This theorem tells us that any variable, including , is effectively not free in , even though is technically free according to the traditional definition of free variable. (Contributed by NM, 13-Jan-2011.) (Revised by NM, 21-Aug-2017.)

Theoremstdpc6 1591 One of the two equality axioms of standard predicate calculus, called reflexivity of equality. (The other one is stdpc7 1653.) Axiom 6 of [Mendelson] p. 95. Mendelson doesn't say why he prepended the redundant quantifier, but it was probably to be compatible with free logic (which is valid in the empty domain). (Contributed by NM, 16-Feb-2005.)

Theoremequcomi 1592 Commutative law for equality. Lemma 7 of [Tarski] p. 69. (Contributed by NM, 5-Aug-1993.)

Theoremequcom 1593 Commutative law for equality. (Contributed by NM, 20-Aug-1993.)

Theoremequcoms 1594 An inference commuting equality in antecedent. Used to eliminate the need for a syllogism. (Contributed by NM, 5-Aug-1993.)

Theoremequtr 1595 A transitive law for equality. (Contributed by NM, 23-Aug-1993.)

Theoremequtrr 1596 A transitive law for equality. Lemma L17 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 23-Aug-1993.)

Theoremequtr2 1597 A transitive law for equality. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremequequ1 1598 An equivalence law for equality. (Contributed by NM, 5-Aug-1993.)

Theoremequequ2 1599 An equivalence law for equality. (Contributed by NM, 5-Aug-1993.)

Theoremelequ1 1600 An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10124
 Copyright terms: Public domain < Previous  Next >