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

Theoremsp 1401 Specialization. Another name for ax-4 1400. (Contributed by NM, 21-May-2008.)

Theoremax-12 1402 Rederive the original version of the axiom from ax-i12 1398. (Contributed by Mario Carneiro, 3-Feb-2015.)

Theoremax12or 1403 Another name for ax-i12 1398. (Contributed by NM, 3-Feb-2015.)

Axiomax-13 1404 Axiom of Equality. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. It substitutes equal variables into the left-hand side of the binary predicate. Axiom scheme C12' in [Megill] p. 448 (p. 16 of the preprint). It is a special case of Axiom B8 (p. 75) of system S2 of [Tarski] p. 77. "Non-logical" means that the predicate is not a primitive of predicate calculus proper but instead is an extension to it. "Binary" means that the predicate has two arguments. In a system of predicate calculus with equality, like ours, equality is not usually considered to be a non-logical predicate. In systems of predicate calculus without equality, it typically would be. (Contributed by NM, 5-Aug-1993.)

Axiomax-14 1405 Axiom of Equality. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. It substitutes equal variables into the right-hand side of the binary predicate. Axiom scheme C13' in [Megill] p. 448 (p. 16 of the preprint). It is a special case of Axiom B8 (p. 75) of system S2 of [Tarski] p. 77. (Contributed by NM, 5-Aug-1993.)

Theoremhbequid 1406 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. (The proof uses only ax-5 1336, ax-8 1395, ax-12 1402, and ax-gen 1338. This shows that this can be proved without ax-9 1424, even though the theorem equid 1589 cannot be. A shorter proof using ax-9 1424 is obtainable from equid 1589 and hbth 1352.) (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 23-Mar-2014.)

Theoremaxi12 1407 Proof that ax-i12 1398 follows from ax-bndl 1399. So that we can track which theorems rely on ax-bndl 1399, proofs should reference ax-i12 1398 rather than this theorem. (Contributed by Jim Kingdon, 17-Aug-2018.) (New usage is discouraged). (Proof modification is discouraged.)

Theoremalequcom 1408 Commutation law for identical variable specifiers. The antecedent and consequent are true when and are substituted with the same variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). (Contributed by NM, 5-Aug-1993.)

Theoremalequcoms 1409 A commutation rule for identical variable specifiers. (Contributed by NM, 5-Aug-1993.)

Theoremnalequcoms 1410 A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 2-Feb-2015.)

Theoremnfr 1411 Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.)

Theoremnfri 1412 Consequence of the definition of not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfrd 1413 Consequence of the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremalimd 1414 Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)

Theoremalrimi 1415 Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)

Theoremnfd 1416 Deduce that is not free in in a context. (Contributed by Mario Carneiro, 24-Sep-2016.)

Theoremnfdh 1417 Deduce that is not free in in a context. (Contributed by Mario Carneiro, 24-Sep-2016.)

Theoremnfrimi 1418 Moving an antecedent outside . (Contributed by Jim Kingdon, 23-Mar-2018.)

1.3.3  Axiom ax-17 - first use of the \$d distinct variable statement

Axiomax-17 1419* Axiom to quantify a variable over a formula in which it does not occur. Axiom C5 in [Megill] p. 444 (p. 11 of the preprint). Also appears as Axiom B6 (p. 75) of system S2 of [Tarski] p. 77 and Axiom C5-1 of [Monk2] p. 113.

(Contributed by NM, 5-Aug-1993.)

Theorema17d 1420* ax-17 1419 with antecedent. (Contributed by NM, 1-Mar-2013.)

Theoremnfv 1421* If is not present in , then is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfvd 1422* nfv 1421 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders such as nfimd 1477. (Contributed by Mario Carneiro, 6-Oct-2016.)

1.3.4  Introduce Axiom of Existence

Axiomax-i9 1423 Axiom of Existence. One of the equality and substitution axioms of predicate calculus with equality. One thing this axiom tells us is that at least one thing exists (although ax-4 1400 and possibly others also tell us that, i.e. they are not valid in the empty domain of a "free logic"). In this form (not requiring that and be distinct) it was used in an axiom system of Tarski (see Axiom B7' in footnote 1 of [KalishMontague] p. 81.) Another name for this theorem is a9e 1586, which has additional remarks. (Contributed by Mario Carneiro, 31-Jan-2015.)

Theoremax-9 1424 Derive ax-9 1424 from ax-i9 1423, the modified version for intuitionistic logic. Although ax-9 1424 does hold intuistionistically, in intuitionistic logic it is weaker than ax-i9 1423. (Contributed by NM, 3-Feb-2015.)

Theoremequidqe 1425 equid 1589 with some quantification and negation without using ax-4 1400 or ax-17 1419. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 27-Feb-2014.)

Theoremax4sp1 1426 A special case of ax-4 1400 without using ax-4 1400 or ax-17 1419. (Contributed by NM, 13-Jan-2011.)

1.3.5  Additional intuitionistic axioms

Axiomax-ial 1427 is not free in . One of the axioms of predicate logic. (Contributed by Mario Carneiro, 31-Jan-2015.)

Axiomax-i5r 1428 Axiom of quantifier collection. (Contributed by Mario Carneiro, 31-Jan-2015.)

1.3.6  Predicate calculus including ax-4, without distinct variables

Theoremspi 1429 Inference rule reversing generalization. (Contributed by NM, 5-Aug-1993.)

Theoremsps 1430 Generalization of antecedent. (Contributed by NM, 5-Aug-1993.)

Theoremspsd 1431 Deduction generalizing antecedent. (Contributed by NM, 17-Aug-1994.)

Theoremnfbidf 1432 An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.)

Theoremhba1 1433 is not free in . Example in Appendix in [Megill] p. 450 (p. 19 of the preprint). Also Lemma 22 of [Monk2] p. 114. (Contributed by NM, 5-Aug-1993.)

Theoremnfa1 1434 is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.)

Theorema5i 1435 Inference generalizing a consequent. (Contributed by NM, 5-Aug-1993.)

Theoremnfnf1 1436 is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremhbim 1437 If is not free in and , it is not free in . (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 3-Mar-2008.) (Revised by Mario Carneiro, 2-Feb-2015.)

Theoremhbor 1438 If is not free in and , it is not free in . (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)

Theoremhban 1439 If is not free in and , it is not free in . (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 2-Feb-2015.)

Theoremhbbi 1440 If is not free in and , it is not free in . (Contributed by NM, 5-Aug-1993.)

Theoremhb3or 1441 If is not free in , , and , it is not free in . (Contributed by NM, 14-Sep-2003.)

Theoremhb3an 1442 If is not free in , , and , it is not free in . (Contributed by NM, 14-Sep-2003.)

Theoremhba2 1443 Lemma 24 of [Monk2] p. 114. (Contributed by NM, 29-May-2008.)

Theoremhbia1 1444 Lemma 23 of [Monk2] p. 114. (Contributed by NM, 29-May-2008.)

Theorem19.3h 1445 A wff may be quantified with a variable not free in it. Theorem 19.3 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 21-May-2007.)

Theorem19.3 1446 A wff may be quantified with a variable not free in it. Theorem 19.3 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)

Theorem19.16 1447 Theorem 19.16 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)

Theorem19.17 1448 Theorem 19.17 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)

Theorem19.21h 1449 Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as " is not free in ." New proofs should use 19.21 1475 instead. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Theorem19.21bi 1450 Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.21bbi 1451 Inference removing double quantifier. (Contributed by NM, 20-Apr-1994.)

Theorem19.27h 1452 Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.27 1453 Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.28h 1454 Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theorem19.28 1455 Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theoremnfan1 1456 A closed form of nfan 1457. (Contributed by Mario Carneiro, 3-Oct-2016.)

Theoremnfan 1457 If is not free in and , it is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 13-Jan-2018.)

Theoremnf3an 1458 If is not free in , , and , it is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnford 1459 If in a context is not free in and , it is not free in . (Contributed by Jim Kingdon, 29-Oct-2019.)

Theoremnfand 1460 If in a context is not free in and , it is not free in . (Contributed by Mario Carneiro, 7-Oct-2016.)

Theoremnf3and 1461 Deduction form of bound-variable hypothesis builder nf3an 1458. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 16-Oct-2016.)

Theoremhbim1 1462 A closed form of hbim 1437. (Contributed by NM, 5-Aug-1993.)

Theoremnfim1 1463 A closed form of nfim 1464. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)

Theoremnfim 1464 If is not free in and , it is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)

Theoremhbimd 1465 Deduction form of bound-variable hypothesis builder hbim 1437. (Contributed by NM, 1-Jan-2002.) (Revised by NM, 2-Feb-2015.)

Theoremnfor 1466 If is not free in and , it is not free in . (Contributed by Jim Kingdon, 11-Mar-2018.)

Theoremhbbid 1467 Deduction form of bound-variable hypothesis builder hbbi 1440. (Contributed by NM, 1-Jan-2002.)

Theoremnfal 1468 If is not free in , it is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfnf 1469 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.)

Theoremnfalt 1470 Closed form of nfal 1468. (Contributed by Jim Kingdon, 11-May-2018.)

Theoremnfa2 1471 Lemma 24 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.)

Theoremnfia1 1472 Lemma 23 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.)

Theorem19.21ht 1473 Closed form of Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 27-May-1997.) (New usage is discouraged.)

Theorem19.21t 1474 Closed form of Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 27-May-1997.)

Theorem19.21 1475 Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as " is not free in ." (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)

Theoremstdpc5 1476 An axiom scheme of standard predicate calculus that emulates Axiom 5 of [Mendelson] p. 69. The hypothesis can be thought of as emulating " is not free in ." With this definition, the meaning of "not free" is less restrictive than the usual textbook definition; for example would not (for us) be free in by nfequid 1590. This theorem scheme can be proved as a metatheorem of Mendelson's axiom system, even though it is slightly stronger than his Axiom 5. (Contributed by NM, 22-Sep-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.)

Theoremnfimd 1477 If in a context is not free in and , it is not free in . (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.)

Theoremaaanh 1478 Rearrange universal quantifiers. (Contributed by NM, 12-Aug-1993.)

Theoremaaan 1479 Rearrange universal quantifiers. (Contributed by NM, 12-Aug-1993.)

Theoremnfbid 1480 If in a context is not free in and , it is not free in . (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 29-Dec-2017.)

Theoremnfbi 1481 If is not free in and , it is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)

1.3.7  The existential quantifier

Theorem19.8a 1482 If a wff is true, it is true for at least one instance. Special case of Theorem 19.8 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)

Theorem19.23bi 1483 Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theoremexlimih 1484 Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Theoremexlimi 1485 Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)

Theoremexlimd2 1486 Deduction from Theorem 19.23 of [Margaris] p. 90. Similar to exlimdh 1487 but with one slightly different hypothesis. (Contributed by Jim Kingdon, 30-Dec-2017.)

Theoremexlimdh 1487 Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 28-Jan-1997.)

Theoremexlimd 1488 Deduction from Theorem 19.9 of [Margaris] p. 89. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof rewritten by Jim Kingdon, 18-Jun-2018.)

Theoremexlimiv 1489* Inference from Theorem 19.23 of [Margaris] p. 90.

This inference, along with our many variants is used to implement a metatheorem called "Rule C" that is given in many logic textbooks. See, for example, Rule C in [Mendelson] p. 81, Rule C in [Margaris] p. 40, or Rule C in Hirst and Hirst's A Primer for Logic and Proof p. 59 (PDF p. 65) at http://www.mathsci.appstate.edu/~jlh/primer/hirst.pdf.

In informal proofs, the statement "Let C be an element such that..." almost always means an implicit application of Rule C.

In essence, Rule C states that if we can prove that some element exists satisfying a wff, i.e. where has free, then we can use C as a hypothesis for the proof where C is a new (ficticious) constant not appearing previously in the proof, nor in any axioms used, nor in the theorem to be proved. The purpose of Rule C is to get rid of the existential quantifier.

We cannot do this in Metamath directly. Instead, we use the original (containing ) as an antecedent for the main part of the proof. We eventually arrive at where is the theorem to be proved and does not contain . Then we apply exlimiv 1489 to arrive at . Finally, we separately prove and detach it with modus ponens ax-mp 7 to arrive at the final theorem . (Contributed by NM, 5-Aug-1993.) (Revised by NM, 25-Jul-2012.)

Theoremexim 1490 Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.)

Theoremeximi 1491 Inference adding existential quantifier to antecedent and consequent. (Contributed by NM, 5-Aug-1993.)

Theorem2eximi 1492 Inference adding 2 existential quantifiers to antecedent and consequent. (Contributed by NM, 3-Feb-2005.)

Theoremeximii 1493 Inference associated with eximi 1491. (Contributed by BJ, 3-Feb-2018.)

Theoremalinexa 1494 A transformation of quantifiers and logical connectives. (Contributed by NM, 19-Aug-1993.)

Theoremexbi 1495 Theorem 19.18 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Theoremexbii 1496 Inference adding existential quantifier to both sides of an equivalence. (Contributed by NM, 24-May-1994.)

Theorem2exbii 1497 Inference adding 2 existential quantifiers to both sides of an equivalence. (Contributed by NM, 16-Mar-1995.)

Theorem3exbii 1498 Inference adding 3 existential quantifiers to both sides of an equivalence. (Contributed by NM, 2-May-1995.)

Theoremexancom 1499 Commutation of conjunction inside an existential quantifier. (Contributed by NM, 18-Aug-1993.)

Theoremalrimdd 1500 Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)

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