Theorem rexcomf 2771

Description: Commutation of restricted quantifiers. (Contributed by Mario Carneiro, 14-Oct-2016.)

Hypotheses

Ref	Expression
ralcomf.1	|- F/_yA
ralcomf.2	|- F/_xB

Assertion

Ref	Expression
rexcomf	|- (E.x e. A E.y e. B ph <-> E.y e. B E.x e. A ph)

Distinct variable group:   x,y

Allowed substitution hints:   ph(x,y)   A(x,y)   B(x,y)

Proof of Theorem rexcomf

Step	Hyp	Ref	Expression
1		ancom 437	. . . . 5 |- ((x e. A /\ y e. B) <-> (y e. B /\ x e. A))
2	1	anbi1i 676	. . . 4 |- (((x e. A /\ y e. B) /\ ph) <-> ((y e. B /\ x e. A) /\ ph))
3	2	2exbii 1583	. . 3 |- (E.xE.y((x e. A /\ y e. B) /\ ph) <-> E.xE.y((y e. B /\ x e. A) /\ ph))
4		excom 1741	. . 3 |- (E.xE.y((y e. B /\ x e. A) /\ ph) <-> E.yE.x((y e. B /\ x e. A) /\ ph))
5	3, 4	bitri 240	. 2 |- (E.xE.y((x e. A /\ y e. B) /\ ph) <-> E.yE.x((y e. B /\ x e. A) /\ ph))
6		ralcomf.1	. . 3 |- F/_yA
7	6	r2exf 2651	. 2 |- (E.x e. A E.y e. B ph <-> E.xE.y((x e. A /\ y e. B) /\ ph))
8		ralcomf.2	. . 3 |- F/_xB
9	8	r2exf 2651	. 2 |- (E.y e. B E.x e. A ph <-> E.yE.x((y e. B /\ x e. A) /\ ph))
10	5, 7, 9	3bitr4i 268	1 |- (E.x e. A E.y e. B ph <-> E.y e. B E.x e. A ph)

Syntax hints:   <-> wb 176   /\ wa 358  E.wex 1541   e. wcel 1710   F/_wnfc 2477  E.wrex 2616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rex 2621

This theorem is referenced by:  rexcom  2773