Theorem rexcomf 2693

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 266	. . . . 5 |- ( ( x e. A /\ y e. B ) <-> ( y e. B /\ x e. A ) )
2	1	anbi1i 458	. . . 4 |- ( ( ( x e. A /\ y e. B ) /\ ph ) <-> ( ( y e. B /\ x e. A ) /\ ph ) )
3	2	2exbii 1652	. . 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 1710	. . 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 184	. 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 2548	. 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 2548	. 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 212	1 |- ( E. x e. A E. y e. B ph <-> E. y e. B E. x e. A ph )

Syntax hints:   /\ wa 104   <-> wb 105   E.wex 1538   e. wcel 2200   F/_wnfc 2359   E.wrex 2509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514

This theorem is referenced by:  rexcom  2695