Theorem rexcomf 2472

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 253	. . . . 5 |- ( ( x e. A /\ y e. B ) <-> ( y e. B /\ x e. A ) )
2	1	anbi1i 431	. . . 4 |- ( ( ( x e. A /\ y e. B ) /\ ph ) <-> ( ( y e. B /\ x e. A ) /\ ph ) )
3	2	2exbii 1497	. . 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 1554	. . 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 173	. 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 2342	. 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 2342	. 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 201	1 |- ( E. x e. A E. y e. B ph <-> E. y e. B E. x e. A ph )

Syntax hints:   /\ wa 97   <-> wb 98   E.wex 1381   e. wcel 1393   F/_wnfc 2165   E.wrex 2307

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312

This theorem is referenced by:  rexcom  2474