Theorem rexcomf 2551

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

Hypotheses

Ref	Expression
ralcomf.1	|- F/_ y A
ralcomf.2	|- F/_ x B

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 264	. . . . 5 |- ( ( x e. A /\ y e. B ) <-> ( y e. B /\ x e. A ) )
2	1	anbi1i 449	. . . 4 |- ( ( ( x e. A /\ y e. B ) /\ ph ) <-> ( ( y e. B /\ x e. A ) /\ ph ) )
3	2	2exbii 1553	. . 3 |- ( E. x E. y ( ( x e. A /\ y e. B ) /\ ph ) <-> E. x E. y ( ( y e. B /\ x e. A ) /\ ph ) )
4		excom 1610	. . 3 |- ( E. x E. y ( ( y e. B /\ x e. A ) /\ ph ) <-> E. y E. x ( ( y e. B /\ x e. A ) /\ ph ) )
5	3, 4	bitri 183	. 2 |- ( E. x E. y ( ( x e. A /\ y e. B ) /\ ph ) <-> E. y E. x ( ( y e. B /\ x e. A ) /\ ph ) )
6		ralcomf.1	. . 3 |- F/_ y A
7	6	r2exf 2412	. 2 |- ( E. x e. A E. y e. B ph <-> E. x E. y ( ( x e. A /\ y e. B ) /\ ph ) )
8		ralcomf.2	. . 3 |- F/_ x B
9	8	r2exf 2412	. 2 |- ( E. y e. B E. x e. A ph <-> E. y E. x ( ( y e. B /\ x e. A ) /\ ph ) )
10	5, 7, 9	3bitr4i 211	1 |- ( E. x e. A E. y e. B ph <-> E. y e. B E. x e. A ph )

Syntax hints:   /\ wa 103   <-> wb 104   E.wex 1436   e. wcel 1448   F/_wnfc 2227   E.wrex 2376

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-nf 1405  df-sb 1704  df-cleq 2093  df-clel 2096  df-nfc 2229  df-rex 2381

This theorem is referenced by:  rexcom  2553