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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.)
(x = y → (x zy z))
 
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.)
(x = y → (z xz y))
 
Theoremhbequid 1403 Bound-variable hypothesis builder for x = x. This theorem tells us that any variable, including x, is effectively not free in x = x, even though x 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.)
(x = xy x = x)
 
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.)
(z z = x (z z = y z(x = yz x = y)))
 
Theoremalequcom 1405 Commutation law for identical variable specifiers. The antecedent and consequent are true when x and y are substituted with the same variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). (Contributed by NM, 5-Aug-1993.)
(x x = yy y = x)
 
Theoremalequcoms 1406 A commutation rule for identical variable specifiers. (Contributed by NM, 5-Aug-1993.)
(x x = yφ)       (y y = xφ)
 
Theoremnalequcoms 1407 A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 2-Feb-2015.)
x x = yφ)       y y = xφ)
 
Theoremnfr 1408 Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.)
(Ⅎxφ → (φxφ))
 
Theoremnfri 1409 Consequence of the definition of not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
xφ       (φxφ)
 
Theoremnfrd 1410 Consequence of the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.)
(φ → Ⅎxψ)       (φ → (ψxψ))
 
Theoremalimd 1411 Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
xφ    &   (φ → (ψχ))       (φ → (xψxχ))
 
Theoremalrimi 1412 Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
xφ    &   (φψ)       (φxψ)
 
Theoremnfd 1413 Deduce that x is not free in ψ in a context. (Contributed by Mario Carneiro, 24-Sep-2016.)
xφ    &   (φ → (ψxψ))       (φ → Ⅎxψ)
 
Theoremnfdh 1414 Deduce that x is not free in ψ in a context. (Contributed by Mario Carneiro, 24-Sep-2016.)
(φxφ)    &   (φ → (ψxψ))       (φ → Ⅎxψ)
 
Theoremnfrimi 1415 Moving an antecedent outside . (Contributed by Jim Kingdon, 23-Mar-2018.)
xφ    &   x(φψ)       (φ → Ⅎxψ)
 
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.)

(φxφ)
 
Theorema17d 1417* ax-17 1416 with antecedent. (Contributed by NM, 1-Mar-2013.)
(φ → (ψxψ))
 
Theoremnfv 1418* If x is not present in φ, then x is not free in φ. (Contributed by Mario Carneiro, 11-Aug-2016.)
xφ
 
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.)
(φ → Ⅎxψ)
 
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 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 1583, which has additional remarks. (Contributed by Mario Carneiro, 31-Jan-2015.)
x x = y
 
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.)
¬ x ¬ x = y
 
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.)
¬ y ¬ x = x
 
Theoremax4sp1 1423 A special case of ax-4 1397 without using ax-4 1397 or ax-17 1416. (Contributed by NM, 13-Jan-2011.)
(y ¬ x = x → ¬ x = x)
 
1.3.5  Additional intuitionistic axioms
 
Axiomax-ial 1424 x is not free in xφ. Axiom 7 of 10 for intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
(xφxxφ)
 
Axiomax-i5r 1425 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 1426 Inference rule reversing generalization. (Contributed by NM, 5-Aug-1993.)
xφ       φ
 
Theoremsps 1427 Generalization of antecedent. (Contributed by NM, 5-Aug-1993.)
(φψ)       (xφψ)
 
Theoremspsd 1428 Deduction generalizing antecedent. (Contributed by NM, 17-Aug-1994.)
(φ → (ψχ))       (φ → (xψχ))
 
Theoremnfbidf 1429 An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.)
xφ    &   (φ → (ψχ))       (φ → (Ⅎxψ ↔ Ⅎxχ))
 
Theoremhba1 1430 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 1431 x is not free in xφ. (Contributed by Mario Carneiro, 11-Aug-2016.)
xxφ
 
Theorema5i 1432 Inference generalizing a consequent. (Contributed by NM, 5-Aug-1993.)
(xφψ)       (xφxψ)
 
Theoremnfnf1 1433 x is not free in xφ. (Contributed by Mario Carneiro, 11-Aug-2016.)
xxφ
 
Theoremhbim 1434 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 1435 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 1436 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 1437 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 1438 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 1439 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 1440 Lemma 24 of [Monk2] p. 114. (Contributed by NM, 29-May-2008.)
(yxφxyxφ)
 
Theoremhbia1 1441 Lemma 23 of [Monk2] p. 114. (Contributed by NM, 29-May-2008.)
((xφxψ) → x(xφxψ))
 
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.)
(φxφ)       (xφφ)
 
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.)
xφ       (xφφ)
 
Theorem19.16 1444 Theorem 19.16 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
xφ       (x(φψ) → (φxψ))
 
Theorem19.17 1445 Theorem 19.17 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
xψ       (x(φψ) → (xφψ))
 
Theorem19.21h 1446 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 1472 instead. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
(φxφ)       (x(φψ) ↔ (φxψ))
 
Theorem19.21bi 1447 Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
(φxψ)       (φψ)
 
Theorem19.21bbi 1448 Inference removing double quantifier. (Contributed by NM, 20-Apr-1994.)
(φxyψ)       (φψ)
 
Theorem19.27h 1449 Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
(ψxψ)       (x(φ ψ) ↔ (xφ ψ))
 
Theorem19.27 1450 Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
xψ       (x(φ ψ) ↔ (xφ ψ))
 
Theorem19.28h 1451 Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
(φxφ)       (x(φ ψ) ↔ (φ xψ))
 
Theorem19.28 1452 Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
xφ       (x(φ ψ) ↔ (φ xψ))
 
Theoremnfan1 1453 A closed form of nfan 1454. (Contributed by Mario Carneiro, 3-Oct-2016.)
xφ    &   (φ → Ⅎxψ)       x(φ ψ)
 
Theoremnfan 1454 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 1455 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 1456 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 1457 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 1458 Deduction form of bound-variable hypothesis builder nf3an 1455. (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 1459 A closed form of hbim 1434. (Contributed by NM, 5-Aug-1993.)
(φxφ)    &   (φ → (ψxψ))       ((φψ) → x(φψ))
 
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.)
xφ    &   (φ → Ⅎxψ)       x(φψ)
 
Theoremnfim 1461 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 1462 Deduction form of bound-variable hypothesis builder hbim 1434. (Contributed by NM, 1-Jan-2002.) (Revised by NM, 2-Feb-2015.)
(φxφ)    &   (φ → (ψxψ))    &   (φ → (χxχ))       (φ → ((ψχ) → x(ψχ)))
 
Theoremnfor 1463 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 1464 Deduction form of bound-variable hypothesis builder hbbi 1437. (Contributed by NM, 1-Jan-2002.)
(φxφ)    &   (φ → (ψxψ))    &   (φ → (χxχ))       (φ → ((ψχ) → x(ψχ)))
 
Theoremnfal 1465 If x is not free in φ, it is not free in yφ. (Contributed by Mario Carneiro, 11-Aug-2016.)
xφ       xyφ
 
Theoremnfnf 1466 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 1467 Closed form of nfal 1465. (Contributed by Jim Kingdon, 11-May-2018.)
(yxφ → Ⅎxyφ)
 
Theoremnfa2 1468 Lemma 24 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.)
xyxφ
 
Theoremnfia1 1469 Lemma 23 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.)
x(xφxψ)
 
Theorem19.21ht 1470 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 1471 Closed form of Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 27-May-1997.)
(Ⅎxφ → (x(φψ) ↔ (φxψ)))
 
Theorem19.21 1472 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 1473 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 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.)
xφ       (x(φψ) → (φxψ))
 
Theoremnfimd 1474 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 1475 Rearrange universal quantifiers. (Contributed by NM, 12-Aug-1993.)
(φyφ)    &   (ψxψ)       (xy(φ ψ) ↔ (xφ yψ))
 
Theoremaaan 1476 Rearrange universal quantifiers. (Contributed by NM, 12-Aug-1993.)
yφ    &   xψ       (xy(φ ψ) ↔ (xφ yψ))
 
Theoremnfbid 1477 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 1478 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 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.)
(φxφ)
 
Theorem19.23bi 1480 Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
(xφψ)       (φψ)
 
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.)
(ψxψ)    &   (φψ)       (xφψ)
 
Theoremexlimi 1482 Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
xψ    &   (φψ)       (xφψ)
 
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.)
(φxφ)    &   (φ → (χxχ))    &   (φ → (ψχ))       (φ → (xψχ))
 
Theoremexlimdh 1484 Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 28-Jan-1997.)
(φxφ)    &   (χxχ)    &   (φ → (ψχ))       (φ → (xψχ))
 
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.)
xφ    &   xχ    &   (φ → (ψχ))       (φ → (xψχ))
 
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 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 1486 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 1487 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 1488 Inference adding existential quantifier to antecedent and consequent. (Contributed by NM, 5-Aug-1993.)
(φψ)       (xφxψ)
 
Theorem2eximi 1489 Inference adding 2 existential quantifiers to antecedent and consequent. (Contributed by NM, 3-Feb-2005.)
(φψ)       (xyφxyψ)
 
Theoremeximii 1490 Inference associated with eximi 1488. (Contributed by BJ, 3-Feb-2018.)
xφ    &   (φψ)       xψ
 
Theoremalinexa 1491 A transformation of quantifiers and logical connectives. (Contributed by NM, 19-Aug-1993.)
(x(φ → ¬ ψ) ↔ ¬ x(φ ψ))
 
Theoremexbi 1492 Theorem 19.18 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
(x(φψ) → (xφxψ))
 
Theoremexbii 1493 Inference adding existential quantifier to both sides of an equivalence. (Contributed by NM, 24-May-1994.)
(φψ)       (xφxψ)
 
Theorem2exbii 1494 Inference adding 2 existential quantifiers to both sides of an equivalence. (Contributed by NM, 16-Mar-1995.)
(φψ)       (xyφxyψ)
 
Theorem3exbii 1495 Inference adding 3 existential quantifiers to both sides of an equivalence. (Contributed by NM, 2-May-1995.)
(φψ)       (xyzφxyzψ)
 
Theoremexancom 1496 Commutation of conjunction inside an existential quantifier. (Contributed by NM, 18-Aug-1993.)
(x(φ ψ) ↔ x(ψ φ))
 
Theoremalrimdd 1497 Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
xφ    &   (φ → Ⅎxψ)    &   (φ → (ψχ))       (φ → (ψxχ))
 
Theoremalrimd 1498 Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
xφ    &   xψ    &   (φ → (ψχ))       (φ → (ψxχ))
 
Theoremeximdh 1499 Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 20-May-1996.)
(φxφ)    &   (φ → (ψχ))       (φ → (xψxχ))
 
Theoremeximd 1500 Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
xφ    &   (φ → (ψχ))       (φ → (xψxχ))
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