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

Theorema17d 1401* ax-17 1400 with antecedent. (Contributed by NM, 1-Mar-2013.)
(φ → (ψxψ))

Theoremnfv 1402* If x is not present in φ, then x is not free in φ. (Contributed by Mario Carneiro, 11-Aug-2016.)
xφ

Theoremnfvd 1403* nfv 1402 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders such as nfimd 1459. (Contributed by Mario Carneiro, 6-Oct-2016.)
(φ → Ⅎxψ)

1.3.4  Introduce Axiom of Existence

Axiomax-i9 1404 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 1381 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 x and y 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 1568, which has additional remarks. (Contributed by Mario Carneiro, 31-Jan-2015.)
x x = y

Theoremax-9 1405 Derive ax-9 1405 from ax-i9 1404, the modified version for intuitionistic logic. Although ax-9 1405 does hold intuistionistically, in intuitionistic logic it is weaker than ax-i9 1404. (Contributed by NM, 3-Feb-2015.)
¬ x ¬ x = y

Theoremequidqe 1406 equid 1571 with some quantification and negation without using ax-4 1381 or ax-17 1400. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 27-Feb-2014.)
¬ y ¬ x = x

TheoremequidqeOLD 1407 Obsolete proof of equidqe 1406 as of 27-Feb-2014. (Contributed by NM, 13-Jan-2011.)
¬ y ¬ x = x

Theoremax4sp1 1408 A special case of ax-4 1381 without using ax-4 1381 or ax-17 1400. (Contributed by NM, 13-Jan-2011.)
(y ¬ x = x → ¬ x = x)

Axiomax-ial 1409 x is not free in xφ. Axiom 7 of 10 for intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
(xφxxφ)

Axiomax-i5r 1410 Axiom of quantifier collection. (Contributed by Mario Carneiro, 31-Jan-2015.)
((xφxψ) → x(xφψ))

1.3.6  Predicate calculus including ax-4, without distinct variables

Theoremspi 1411 Inference rule reversing generalization. (Contributed by NM, 5-Aug-1993.)
xφ       φ

Theoremsps 1412 Generalization of antecedent. (Contributed by NM, 5-Aug-1993.)
(φψ)       (xφψ)

Theoremspsd 1413 Deduction generalizing antecedent. (Contributed by NM, 17-Aug-1994.)
(φ → (ψχ))       (φ → (xψχ))

Theoremnfbidf 1414 An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.)
xφ    &   (φ → (ψχ))       (φ → (Ⅎxψ ↔ Ⅎxχ))

Theoremhba1 1415 x is not free in xφ. 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.)
(xφxxφ)

Theoremnfa1 1416 x is not free in xφ. (Contributed by Mario Carneiro, 11-Aug-2016.)
xxφ

Theorema5i 1417 Inference generalizing a consequent. (Contributed by NM, 5-Aug-1993.)
(xφψ)       (xφxψ)

Theoremnfnf1 1418 x is not free in xφ. (Contributed by Mario Carneiro, 11-Aug-2016.)
xxφ

Theoremhbim 1419 If x is not free in φ and ψ, it is not free in (φψ). (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 3-Mar-2008.) (Revised by Mario Carneiro, 2-Feb-2015.)
(φxφ)    &   (ψxψ)       ((φψ) → x(φψ))

Theoremhbor 1420 If x is not free in φ and ψ, it is not free in (φ ψ). (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)
(φxφ)    &   (ψxψ)       ((φ ψ) → x(φ ψ))

Theoremhban 1421 If x is not free in φ and ψ, it is not free in (φ ψ). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 2-Feb-2015.)
(φxφ)    &   (ψxψ)       ((φ ψ) → x(φ ψ))

Theoremhbbi 1422 If x is not free in φ and ψ, it is not free in (φψ). (Contributed by NM, 5-Aug-1993.)
(φxφ)    &   (ψxψ)       ((φψ) → x(φψ))

Theoremhb3or 1423 If x is not free in φ, ψ, and χ, it is not free in (φ ψ χ). (Contributed by NM, 14-Sep-2003.)
(φxφ)    &   (ψxψ)    &   (χxχ)       ((φ ψ χ) → x(φ ψ χ))

Theoremhb3an 1424 If x is not free in φ, ψ, and χ, it is not free in (φ ψ χ). (Contributed by NM, 14-Sep-2003.)
(φxφ)    &   (ψxψ)    &   (χxχ)       ((φ ψ χ) → x(φ ψ χ))

Theoremhba2 1425 Lemma 24 of [Monk2] p. 114. (Contributed by NM, 29-May-2008.)
(yxφxyxφ)

Theoremhbia1 1426 Lemma 23 of [Monk2] p. 114. (Contributed by NM, 29-May-2008.)
((xφxψ) → x(xφxψ))

Theorem19.3h 1427 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.)
(φxφ)       (xφφ)

Theorem19.3 1428 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.)
xφ       (xφφ)

Theorem19.16 1429 Theorem 19.16 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
xφ       (x(φψ) → (φxψ))

Theorem19.17 1430 Theorem 19.17 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
xψ       (x(φψ) → (xφψ))

Theorem19.21h 1431 Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "x is not free in φ." New proofs should use 19.21 1457 instead. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
(φxφ)       (x(φψ) ↔ (φxψ))

Theorem19.21bi 1432 Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
(φxψ)       (φψ)

Theorem19.21bbi 1433 Inference removing double quantifier. (Contributed by NM, 20-Apr-1994.)
(φxyψ)       (φψ)

Theorem19.27h 1434 Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
(ψxψ)       (x(φ ψ) ↔ (xφ ψ))

Theorem19.27 1435 Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
xψ       (x(φ ψ) ↔ (xφ ψ))

Theorem19.28h 1436 Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
(φxφ)       (x(φ ψ) ↔ (φ xψ))

Theorem19.28 1437 Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
xφ       (x(φ ψ) ↔ (φ xψ))

Theoremnfan1 1438 A closed form of nfan 1439. (Contributed by Mario Carneiro, 3-Oct-2016.)
xφ    &   (φ → Ⅎxψ)       x(φ ψ)

Theoremnfan 1439 If x is not free in φ and ψ, it is not free in (φ ψ). (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 13-Jan-2018.)
xφ    &   xψ       x(φ ψ)

Theoremnf3an 1440 If x is not free in φ, ψ, and χ, it is not free in (φ ψ χ). (Contributed by Mario Carneiro, 11-Aug-2016.)
xφ    &   xψ    &   xχ       x(φ ψ χ)

Theoremnford 1441 If in a context x is not free in ψ and χ, it is not free in (ψ χ). (Contributed by Jim Kingdon, 29-Oct-2019.)
(φ → Ⅎxψ)    &   (φ → Ⅎxχ)       (φ → Ⅎx(ψ χ))

Theoremnfand 1442 If in a context x is not free in ψ and χ, it is not free in (ψ χ). (Contributed by Mario Carneiro, 7-Oct-2016.)
(φ → Ⅎxψ)    &   (φ → Ⅎxχ)       (φ → Ⅎx(ψ χ))

Theoremnf3and 1443 Deduction form of bound-variable hypothesis builder nf3an 1440. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 16-Oct-2016.)
(φ → Ⅎxψ)    &   (φ → Ⅎxχ)    &   (φ → Ⅎxθ)       (φ → Ⅎx(ψ χ θ))

Theoremhbim1 1444 A closed form of hbim 1419. (Contributed by NM, 5-Aug-1993.)
(φxφ)    &   (φ → (ψxψ))       ((φψ) → x(φψ))

Theoremnfim1 1445 A closed form of nfim 1446. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
xφ    &   (φ → Ⅎxψ)       x(φψ)

Theoremnfim 1446 If x is not free in φ and ψ, it is not free in (φψ). (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
xφ    &   xψ       x(φψ)

Theoremhbimd 1447 Deduction form of bound-variable hypothesis builder hbim 1419. (Contributed by NM, 1-Jan-2002.) (Revised by NM, 2-Feb-2015.)
(φxφ)    &   (φ → (ψxψ))    &   (φ → (χxχ))       (φ → ((ψχ) → x(ψχ)))

Theoremnfor 1448 If x is not free in φ and ψ, it is not free in (φ ψ). (Contributed by Jim Kingdon, 11-Mar-2018.)
xφ    &   xψ       x(φ ψ)

Theoremhbbid 1449 Deduction form of bound-variable hypothesis builder hbbi 1422. (Contributed by NM, 1-Jan-2002.)
(φxφ)    &   (φ → (ψxψ))    &   (φ → (χxχ))       (φ → ((ψχ) → x(ψχ)))

Theoremnfal 1450 If x is not free in φ, it is not free in yφ. (Contributed by Mario Carneiro, 11-Aug-2016.)
xφ       xyφ

Theoremnfnf 1451 If x is not free in φ, it is not free in yφ. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.)
xφ       xyφ

Theoremnfalt 1452 Closed form of nfal 1450. (Contributed by Jim Kingdon, 11-May-2018.)
(yxφ → Ⅎxyφ)

Theoremnfa2 1453 Lemma 24 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.)
xyxφ

Theoremnfia1 1454 Lemma 23 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.)
x(xφxψ)

Theorem19.21ht 1455 Closed form of Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 27-May-1997.) (New usage is discouraged.)
(x(φxφ) → (x(φψ) ↔ (φxψ)))

Theorem19.21t 1456 Closed form of Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 27-May-1997.)
(Ⅎxφ → (x(φψ) ↔ (φxψ)))

Theorem19.21 1457 Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "x is not free in φ." (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
xφ       (x(φψ) ↔ (φxψ))

Theoremstdpc5 1458 An axiom scheme of standard predicate calculus that emulates Axiom 5 of [Mendelson] p. 69. The hypothesis xφ can be thought of as emulating "x is not free in φ." With this definition, the meaning of "not free" is less restrictive than the usual textbook definition; for example x would not (for us) be free in x = x by nfequid 1572. 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.)
xφ       (x(φψ) → (φxψ))

Theoremnfimd 1459 If in a context x is not free in ψ and χ, it is not free in (ψχ). (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.)
(φ → Ⅎxψ)    &   (φ → Ⅎxχ)       (φ → Ⅎx(ψχ))

Theoremaaanh 1460 Rearrange universal quantifiers. (Contributed by NM, 12-Aug-1993.)
(φyφ)    &   (ψxψ)       (xy(φ ψ) ↔ (xφ yψ))

Theoremaaan 1461 Rearrange universal quantifiers. (Contributed by NM, 12-Aug-1993.)
yφ    &   xψ       (xy(φ ψ) ↔ (xφ yψ))

Theoremnfbid 1462 If in a context x is not free in ψ and χ, it is not free in (ψχ). (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 29-Dec-2017.)
(φ → Ⅎxψ)    &   (φ → Ⅎxχ)       (φ → Ⅎx(ψχ))

Theoremnfbi 1463 If x is not free in φ and ψ, it is not free in (φψ). (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
xφ    &   xψ       x(φψ)

1.3.7  The existential quantifier

Theorem19.8a 1464 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.)
(φxφ)

Theorem19.23bi 1465 Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
(xφψ)       (φψ)

Theoremexlimih 1466 Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)
(ψxψ)    &   (φψ)       (xφψ)

Theoremexlimi 1467 Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
xψ    &   (φψ)       (xφψ)

Theoremexlimd2 1468 Deduction from Theorem 19.23 of [Margaris] p. 90. Similar to exlimdh 1469 but with one slightly different hypothesis. (Contributed by Jim Kingdon, 30-Dec-2017.)
(φxφ)    &   (φ → (χxχ))    &   (φ → (ψχ))       (φ → (xψχ))

Theoremexlimdh 1469 Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 28-Jan-1997.)
(φxφ)    &   (χxχ)    &   (φ → (ψχ))       (φ → (xψχ))

Theoremexlimd 1470 Deduction from Theorem 19.9 of [Margaris] p. 89. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof rewritten by Jim Kingdon, 18-Jun-2018.)
xφ    &   xχ    &   (φ → (ψχ))       (φ → (xψχ))

Theoremexlimiv 1471* 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 x exists satisfying a wff, i.e. xφ(x) where φ(x) has x free, then we can use φ( 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 φ (containing x) as an antecedent for the main part of the proof. We eventually arrive at (φψ) where ψ is the theorem to be proved and does not contain x. Then we apply exlimiv 1471 to arrive at (xφψ). Finally, we separately prove xφ and detach it with modus ponens ax-mp 7 to arrive at the final theorem ψ. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 25-Jul-2012.)

(φψ)       (xφψ)

Theoremexim 1472 Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.)
(x(φψ) → (xφxψ))

Theoremeximi 1473 Inference adding existential quantifier to antecedent and consequent. (Contributed by NM, 5-Aug-1993.)
(φψ)       (xφxψ)

Theorem2eximi 1474 Inference adding 2 existential quantifiers to antecedent and consequent. (Contributed by NM, 3-Feb-2005.)
(φψ)       (xyφxyψ)

Theoremeximii 1475 Inference associated with eximi 1473. (Contributed by BJ, 3-Feb-2018.)
xφ    &   (φψ)       xψ

Theoremalinexa 1476 A transformation of quantifiers and logical connectives. (Contributed by NM, 19-Aug-1993.)
(x(φ → ¬ ψ) ↔ ¬ x(φ ψ))

Theoremexbi 1477 Theorem 19.18 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
(x(φψ) → (xφxψ))

Theoremexbii 1478 Inference adding existential quantifier to both sides of an equivalence. (Contributed by NM, 24-May-1994.)
(φψ)       (xφxψ)

Theorem2exbii 1479 Inference adding 2 existential quantifiers to both sides of an equivalence. (Contributed by NM, 16-Mar-1995.)
(φψ)       (xyφxyψ)

Theorem3exbii 1480 Inference adding 3 existential quantifiers to both sides of an equivalence. (Contributed by NM, 2-May-1995.)
(φψ)       (xyzφxyzψ)

Theoremexancom 1481 Commutation of conjunction inside an existential quantifier. (Contributed by NM, 18-Aug-1993.)
(x(φ ψ) ↔ x(ψ φ))

Theoremalrimdd 1482 Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
xφ    &   (φ → Ⅎxψ)    &   (φ → (ψχ))       (φ → (ψxχ))

Theoremalrimd 1483 Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
xφ    &   xψ    &   (φ → (ψχ))       (φ → (ψxχ))

Theoremeximdh 1484 Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 20-May-1996.)
(φxφ)    &   (φ → (ψχ))       (φ → (xψxχ))

Theoremeximd 1485 Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
xφ    &   (φ → (ψχ))       (φ → (xψxχ))

Theoremnexd 1486 Deduction for generalization rule for negated wff. (Contributed by NM, 2-Jan-2002.)
(φxφ)    &   (φ → ¬ ψ)       (φ → ¬ xψ)

Theoremexbidh 1487 Formula-building rule for existential quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.)
(φxφ)    &   (φ → (ψχ))       (φ → (xψxχ))

Theoremalbid 1488 Formula-building rule for universal quantifier (deduction rule). (Contributed by Mario Carneiro, 24-Sep-2016.)
xφ    &   (φ → (ψχ))       (φ → (xψxχ))

Theoremexbid 1489 Formula-building rule for existential quantifier (deduction rule). (Contributed by Mario Carneiro, 24-Sep-2016.)
xφ    &   (φ → (ψχ))       (φ → (xψxχ))

Theoremexsimpl 1490 Simplification of an existentially quantified conjunction. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(x(φ ψ) → xφ)

Theoremexsimpr 1491 Simplification of an existentially quantified conjunction. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(x(φ ψ) → xψ)

Theoremalexdc 1492 Theorem 19.6 of [Margaris] p. 89, given a decidability condition. The forward direction holds for all propositions, as seen at alexim 1518. (Contributed by Jim Kingdon, 2-Jun-2018.)
(xDECID φ → (xφ ↔ ¬ x ¬ φ))

Theorem19.29 1493 Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((xφ xψ) → x(φ ψ))

Theorem19.29r 1494 Variation of Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.)
((xφ xψ) → x(φ ψ))

Theorem19.29r2 1495 Variation of Theorem 19.29 of [Margaris] p. 90 with double quantification. (Contributed by NM, 3-Feb-2005.)
((xyφ xyψ) → xy(φ ψ))

Theorem19.29x 1496 Variation of Theorem 19.29 of [Margaris] p. 90 with mixed quantification. (Contributed by NM, 11-Feb-2005.)
((xyφ xyψ) → xy(φ ψ))

Theorem19.35-1 1497 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.)
(x(φψ) → (xφxψ))

Theorem19.35i 1498 Inference from Theorem 19.35 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)
x(φψ)       (xφxψ)

Theorem19.25 1499 Theorem 19.25 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)
(yx(φψ) → (yxφyxψ))

Theorem19.30dc 1500 Theorem 19.30 of [Margaris] p. 90, with an additional decidability condition. (Contributed by Jim Kingdon, 21-Jul-2018.)
(DECID xψ → (x(φ ψ) → (xφ xψ)))

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