Theorem reean 2557

Description: Rearrange existential quantifiers. (Contributed by NM, 27-Oct-2010.) (Proof shortened by Andrew Salmon, 30-May-2011.)

Hypotheses

Ref	Expression
reean.1	|- F/ y ph
reean.2	|- F/ x ps

Assertion

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

Distinct variable groups:   y, A   x, B   x, y

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

Proof of Theorem reean

Step	Hyp	Ref	Expression
1		an4 556	. . . 4 |- ( ( ( x e. A /\ y e. B ) /\ ( ph /\ ps ) ) <-> ( ( x e. A /\ ph ) /\ ( y e. B /\ ps ) ) )
2	1	2exbii 1553	. . 3 |- ( E. x E. y ( ( x e. A /\ y e. B ) /\ ( ph /\ ps ) ) <-> E. x E. y ( ( x e. A /\ ph ) /\ ( y e. B /\ ps ) ) )
3		nfv 1476	. . . . 5 |- F/ y x e. A
4		reean.1	. . . . 5 |- F/ y ph
5	3, 4	nfan 1512	. . . 4 |- F/ y ( x e. A /\ ph )
6		nfv 1476	. . . . 5 |- F/ x y e. B
7		reean.2	. . . . 5 |- F/ x ps
8	6, 7	nfan 1512	. . . 4 |- F/ x ( y e. B /\ ps )
9	5, 8	eean 1866	. . 3 |- ( E. x E. y ( ( x e. A /\ ph ) /\ ( y e. B /\ ps ) ) <-> ( E. x ( x e. A /\ ph ) /\ E. y ( y e. B /\ ps ) ) )
10	2, 9	bitri 183	. 2 |- ( E. x E. y ( ( x e. A /\ y e. B ) /\ ( ph /\ ps ) ) <-> ( E. x ( x e. A /\ ph ) /\ E. y ( y e. B /\ ps ) ) )
11		r2ex 2414	. 2 |- ( E. x e. A E. y e. B ( ph /\ ps ) <-> E. x E. y ( ( x e. A /\ y e. B ) /\ ( ph /\ ps ) ) )
12		df-rex 2381	. . 3 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
13		df-rex 2381	. . 3 |- ( E. y e. B ps <-> E. y ( y e. B /\ ps ) )
14	12, 13	anbi12i 451	. 2 |- ( ( E. x e. A ph /\ E. y e. B ps ) <-> ( E. x ( x e. A /\ ph ) /\ E. y ( y e. B /\ ps ) ) )
15	10, 11, 14	3bitr4i 211	1 |- ( E. x e. A E. y e. B ( ph /\ ps ) <-> ( E. x e. A ph /\ E. y e. B ps ) )

Syntax hints:   /\ wa 103   <-> wb 104   F/wnf 1404   E.wex 1436   e. wcel 1448   E.wrex 2376

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 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-nf 1405  df-sb 1704  df-cleq 2093  df-clel 2096  df-nfc 2229  df-rex 2381

This theorem is referenced by:  reeanv  2558