Theorem rexcomf 2668

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 1629	. . 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 1687	. . 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 2524	. 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 2524	. 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 1515   e. wcel 2176   F/_wnfc 2335   E.wrex 2485

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490

This theorem is referenced by:  rexcom  2670