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Theorem List for Intuitionistic Logic Explorer - 1401-1500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Axiomax-13 1401 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 1402 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 1403 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 1333, ax-8 1392, ax-12 1399, and ax-gen 1335. This shows that this can be proved without ax-9 1421, even though the theorem equid 1586 cannot be. A shorter proof using ax-9 1421 is obtainable from equid 1586 and hbth 1349.) (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 23-Mar-2014.)
 
Theoremaxi12 1404 Proof that ax-i12 1395 follows from ax-bnd 1396. So that we can track which theorems rely on ax-bnd 1396, proofs should reference ax-i12 1395 rather than this theorem. (Contributed by Jim Kingdon, 17-Aug-2018.) (New usage is discouraged). (Proof modification is discouraged.)
 
Theoremalequcom 1405 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 1406 A commutation rule for identical variable specifiers. (Contributed by NM, 5-Aug-1993.)
   =>   
 
Theoremnalequcoms 1407 A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 2-Feb-2015.)
   =>   
 
Theoremnfr 1408 Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.)
 F/
 
Theoremnfri 1409 Consequence of the definition of not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)

 F/   =>   
 
Theoremnfrd 1410 Consequence of the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.)
 F/   =>   
 
Theoremalimd 1411 Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)

 F/   &       =>   
 
Theoremalrimi 1412 Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)

 F/   &       =>   
 
Theoremnfd 1413 Deduce that is not free in in a context. (Contributed by Mario Carneiro, 24-Sep-2016.)

 F/   &       =>     F/
 
Theoremnfdh 1414 Deduce that is not free in in a context. (Contributed by Mario Carneiro, 24-Sep-2016.)
   &       =>     F/
 
Theoremnfrimi 1415 Moving an antecedent outside  F/. (Contributed by Jim Kingdon, 23-Mar-2018.)

 F/   &     F/    =>     F/
 
1.3.3  Axiom ax-17 - first use of the $d distinct variable statement
 
Axiomax-17 1416* 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 1417* ax-17 1416 with antecedent. (Contributed by NM, 1-Mar-2013.)
 
Theoremnfv 1418* If is not present in , then is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.)

 F/
 
Theoremnfvd 1419* nfv 1418 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders such as nfimd 1474. (Contributed by Mario Carneiro, 6-Oct-2016.)
 F/
 
1.3.4  Introduce Axiom of Existence
 
Axiomax-i9 1420 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 1397 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 1583, which has additional remarks. (Contributed by Mario Carneiro, 31-Jan-2015.)
 
Theoremax-9 1421 Derive ax-9 1421 from ax-i9 1420, the modified version for intuitionistic logic. Although ax-9 1421 does hold intuistionistically, in intuitionistic logic it is weaker than ax-i9 1420. (Contributed by NM, 3-Feb-2015.)
 
Theoremequidqe 1422 equid 1586 with some quantification and negation without using ax-4 1397 or ax-17 1416. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 27-Feb-2014.)
 
Theoremax4sp1 1423 A special case of ax-4 1397 without using ax-4 1397 or ax-17 1416. (Contributed by NM, 13-Jan-2011.)
 
1.3.5  Additional intuitionistic axioms
 
Axiomax-ial 1424 is not free in . Axiom 7 of 10 for intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
 
Axiomax-i5r 1425 Axiom of quantifier collection. (Contributed by Mario Carneiro, 31-Jan-2015.)
 
1.3.6  Predicate calculus including ax-4, without distinct variables
 
Theoremspi 1426 Inference rule reversing generalization. (Contributed by NM, 5-Aug-1993.)
   =>   
 
Theoremsps 1427 Generalization of antecedent. (Contributed by NM, 5-Aug-1993.)
   =>   
 
Theoremspsd 1428 Deduction generalizing antecedent. (Contributed by NM, 17-Aug-1994.)
   =>   
 
Theoremnfbidf 1429 An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.)

 F/   &       =>     F/  F/
 
Theoremhba1 1430 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 1431 is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.)

 F/
 
Theorema5i 1432 Inference generalizing a consequent. (Contributed by NM, 5-Aug-1993.)
   =>   
 
Theoremnfnf1 1433 is not free in  F/. (Contributed by Mario Carneiro, 11-Aug-2016.)

 F/ F/
 
Theoremhbim 1434 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 1435 If is not free in and , it is not free in . (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)
   &       =>   
 
Theoremhban 1436 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 1437 If is not free in and , it is not free in . (Contributed by NM, 5-Aug-1993.)
   &       =>   
 
Theoremhb3or 1438 If is not free in , , and , it is not free in . (Contributed by NM, 14-Sep-2003.)
   &       &       =>   
 
Theoremhb3an 1439 If is not free in , , and , it is not free in . (Contributed by NM, 14-Sep-2003.)
   &       &       =>   
 
Theoremhba2 1440 Lemma 24 of [Monk2] p. 114. (Contributed by NM, 29-May-2008.)
 
Theoremhbia1 1441 Lemma 23 of [Monk2] p. 114. (Contributed by NM, 29-May-2008.)
 
Theorem19.3h 1442 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 1443 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.)

 F/   =>   
 
Theorem19.16 1444 Theorem 19.16 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)

 F/   =>   
 
Theorem19.17 1445 Theorem 19.17 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)

 F/   =>   
 
Theorem19.21h 1446 Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as " is not free in ." New proofs should use 19.21 1472 instead. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
   =>   
 
Theorem19.21bi 1447 Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
   =>   
 
Theorem19.21bbi 1448 Inference removing double quantifier. (Contributed by NM, 20-Apr-1994.)
   =>   
 
Theorem19.27h 1449 Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
   =>   
 
Theorem19.27 1450 Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

 F/   =>   
 
Theorem19.28h 1451 Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
   =>   
 
Theorem19.28 1452 Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

 F/   =>   
 
Theoremnfan1 1453 A closed form of nfan 1454. (Contributed by Mario Carneiro, 3-Oct-2016.)

 F/   &     F/   =>    
 F/
 
Theoremnfan 1454 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.)

 F/   &     F/   =>     F/
 
Theoremnf3an 1455 If is not free in , , and , it is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.)

 F/   &     F/   &     F/   =>    
 F/
 
Theoremnford 1456 If in a context is not free in and , it is not free in . (Contributed by Jim Kingdon, 29-Oct-2019.)
 F/   &     F/   =>     F/
 
Theoremnfand 1457 If in a context is not free in and , it is not free in . (Contributed by Mario Carneiro, 7-Oct-2016.)
 F/   &     F/   =>     F/
 
Theoremnf3and 1458 Deduction form of bound-variable hypothesis builder nf3an 1455. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 16-Oct-2016.)
 F/   &     F/   &     F/   =>     F/
 
Theoremhbim1 1459 A closed form of hbim 1434. (Contributed by NM, 5-Aug-1993.)
   &       =>   
 
Theoremnfim1 1460 A closed form of nfim 1461. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)

 F/   &     F/   =>    
 F/
 
Theoremnfim 1461 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.)

 F/   &     F/   =>     F/
 
Theoremhbimd 1462 Deduction form of bound-variable hypothesis builder hbim 1434. (Contributed by NM, 1-Jan-2002.) (Revised by NM, 2-Feb-2015.)
   &       &       =>   
 
Theoremnfor 1463 If is not free in and , it is not free in . (Contributed by Jim Kingdon, 11-Mar-2018.)

 F/   &     F/   =>     F/
 
Theoremhbbid 1464 Deduction form of bound-variable hypothesis builder hbbi 1437. (Contributed by NM, 1-Jan-2002.)
   &       &       =>   
 
Theoremnfal 1465 If is not free in , it is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.)

 F/   =>    
 F/
 
Theoremnfnf 1466 If is not free in , it is not free in  F/. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.)

 F/   =>    
 F/ F/
 
Theoremnfalt 1467 Closed form of nfal 1465. (Contributed by Jim Kingdon, 11-May-2018.)
 F/  F/
 
Theoremnfa2 1468 Lemma 24 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.)

 F/
 
Theoremnfia1 1469 Lemma 23 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.)

 F/
 
Theorem19.21ht 1470 Closed form of Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 27-May-1997.) (New usage is discouraged.)
 
Theorem19.21t 1471 Closed form of Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 27-May-1997.)
 F/
 
Theorem19.21 1472 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.)

 F/   =>   
 
Theoremstdpc5 1473 An axiom scheme of standard predicate calculus that emulates Axiom 5 of [Mendelson] p. 69. The hypothesis  F/ 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 1587. 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.)

 F/   =>   
 
Theoremnfimd 1474 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.)
 F/   &     F/   =>     F/
 
Theoremaaanh 1475 Rearrange universal quantifiers. (Contributed by NM, 12-Aug-1993.)
   &       =>   
 
Theoremaaan 1476 Rearrange universal quantifiers. (Contributed by NM, 12-Aug-1993.)

 F/   &     F/   =>   
 
Theoremnfbid 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, 29-Dec-2017.)
 F/   &     F/   =>     F/
 
Theoremnfbi 1478 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.)

 F/   &     F/   =>     F/
 
1.3.7  The existential quantifier
 
Theorem19.8a 1479 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 1480 Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
   =>   
 
Theoremexlimih 1481 Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)
   &       =>   
 
Theoremexlimi 1482 Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)

 F/   &       =>   
 
Theoremexlimd2 1483 Deduction from Theorem 19.23 of [Margaris] p. 90. Similar to exlimdh 1484 but with one slightly different hypothesis. (Contributed by Jim Kingdon, 30-Dec-2017.)
   &       &       =>   
 
Theoremexlimdh 1484 Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 28-Jan-1997.)
   &       &       =>   
 
Theoremexlimd 1485 Deduction from Theorem 19.9 of [Margaris] p. 89. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof rewritten by Jim Kingdon, 18-Jun-2018.)

 F/   &     F/   &       =>   
 
Theoremexlimiv 1486* 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 1486 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 1487 Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.)
 
Theoremeximi 1488 Inference adding existential quantifier to antecedent and consequent. (Contributed by NM, 5-Aug-1993.)
   =>   
 
Theorem2eximi 1489 Inference adding 2 existential quantifiers to antecedent and consequent. (Contributed by NM, 3-Feb-2005.)
   =>   
 
Theoremeximii 1490 Inference associated with eximi 1488. (Contributed by BJ, 3-Feb-2018.)
   &       =>   
 
Theoremalinexa 1491 A transformation of quantifiers and logical connectives. (Contributed by NM, 19-Aug-1993.)
 
Theoremexbi 1492 Theorem 19.18 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 
Theoremexbii 1493 Inference adding existential quantifier to both sides of an equivalence. (Contributed by NM, 24-May-1994.)
   =>   
 
Theorem2exbii 1494 Inference adding 2 existential quantifiers to both sides of an equivalence. (Contributed by NM, 16-Mar-1995.)
   =>   
 
Theorem3exbii 1495 Inference adding 3 existential quantifiers to both sides of an equivalence. (Contributed by NM, 2-May-1995.)
   =>   
 
Theoremexancom 1496 Commutation of conjunction inside an existential quantifier. (Contributed by NM, 18-Aug-1993.)
 
Theoremalrimdd 1497 Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)

 F/   &     F/   &       =>   
 
Theoremalrimd 1498 Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)

 F/   &     F/   &       =>   
 
Theoremeximdh 1499 Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 20-May-1996.)
   &       =>   
 
Theoremeximd 1500 Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)

 F/   &       =>   
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