Theorem r2exf 2495

Description: Double restricted existential quantification. (Contributed by Mario Carneiro, 14-Oct-2016.)

Hypothesis

Ref	Expression
r2alf.1	|- F/_ y A

Assertion

Ref	Expression
r2exf	|- ( E. x e. A E. y e. B ph <-> E. x E. y ( ( x e. A /\ y e. B ) /\ ph ) )

Distinct variable group:   x, y

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

Proof of Theorem r2exf

Step	Hyp	Ref	Expression
1		df-rex 2461	. 2 |- ( E. x e. A E. y e. B ph <-> E. x ( x e. A /\ E. y e. B ph ) )
2		r2alf.1	. . . . . 6 |- F/_ y A
3	2	nfcri 2313	. . . . 5 |- F/ y x e. A
4	3	19.42 1688	. . . 4 |- ( E. y ( x e. A /\ ( y e. B /\ ph ) ) <-> ( x e. A /\ E. y ( y e. B /\ ph ) ) )
5		anass 401	. . . . 5 |- ( ( ( x e. A /\ y e. B ) /\ ph ) <-> ( x e. A /\ ( y e. B /\ ph ) ) )
6	5	exbii 1605	. . . 4 |- ( E. y ( ( x e. A /\ y e. B ) /\ ph ) <-> E. y ( x e. A /\ ( y e. B /\ ph ) ) )
7		df-rex 2461	. . . . 5 |- ( E. y e. B ph <-> E. y ( y e. B /\ ph ) )
8	7	anbi2i 457	. . . 4 |- ( ( x e. A /\ E. y e. B ph ) <-> ( x e. A /\ E. y ( y e. B /\ ph ) ) )
9	4, 6, 8	3bitr4i 212	. . 3 |- ( E. y ( ( x e. A /\ y e. B ) /\ ph ) <-> ( x e. A /\ E. y e. B ph ) )
10	9	exbii 1605	. 2 |- ( E. x E. y ( ( x e. A /\ y e. B ) /\ ph ) <-> E. x ( x e. A /\ E. y e. B ph ) )
11	1, 10	bitr4i 187	1 |- ( E. x e. A E. y e. B ph <-> E. x E. y ( ( x e. A /\ y e. B ) /\ ph ) )

Syntax hints:   /\ wa 104   <-> wb 105   E.wex 1492   e. wcel 2148   F/_wnfc 2306   E.wrex 2456

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461

This theorem is referenced by:  r2ex  2497  rexcomf  2639