Theorem r2exf 2407

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 2376	. 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 2229	. . . . 5 |- F/ y x e. A
4	3	19.42 1630	. . . 4 |- ( E. y ( x e. A /\ ( y e. B /\ ph ) ) <-> ( x e. A /\ E. y ( y e. B /\ ph ) ) )
5		anass 394	. . . . 5 |- ( ( ( x e. A /\ y e. B ) /\ ph ) <-> ( x e. A /\ ( y e. B /\ ph ) ) )
6	5	exbii 1548	. . . 4 |- ( E. y ( ( x e. A /\ y e. B ) /\ ph ) <-> E. y ( x e. A /\ ( y e. B /\ ph ) ) )
7		df-rex 2376	. . . . 5 |- ( E. y e. B ph <-> E. y ( y e. B /\ ph ) )
8	7	anbi2i 446	. . . 4 |- ( ( x e. A /\ E. y e. B ph ) <-> ( x e. A /\ E. y ( y e. B /\ ph ) ) )
9	4, 6, 8	3bitr4i 211	. . 3 |- ( E. y ( ( x e. A /\ y e. B ) /\ ph ) <-> ( x e. A /\ E. y e. B ph ) )
10	9	exbii 1548	. 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 186	1 |- ( E. x e. A E. y e. B ph <-> E. x E. y ( ( x e. A /\ y e. B ) /\ ph ) )

Syntax hints:   /\ wa 103   <-> wb 104   E.wex 1433   e. wcel 1445   F/_wnfc 2222   E.wrex 2371

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 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-nf 1402  df-sb 1700  df-cleq 2088  df-clel 2091  df-nfc 2224  df-rex 2376

This theorem is referenced by:  r2ex  2409  rexcomf  2543