Theorem reean 2777

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

Hypotheses

Ref	Expression
reean.1	|- F/yph
reean.2	|- F/xps

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 797	. . . 4 |- (((x e. A /\ y e. B) /\ (ph /\ ps)) <-> ((x e. A /\ ph) /\ (y e. B /\ ps)))
2	1	2exbii 1583	. . 3 |- (E.xE.y((x e. A /\ y e. B) /\ (ph /\ ps)) <-> E.xE.y((x e. A /\ ph) /\ (y e. B /\ ps)))
3		nfv 1619	. . . . 5 |- F/y x e. A
4		reean.1	. . . . 5 |- F/yph
5	3, 4	nfan 1824	. . . 4 |- F/y(x e. A /\ ph)
6		nfv 1619	. . . . 5 |- F/x y e. B
7		reean.2	. . . . 5 |- F/xps
8	6, 7	nfan 1824	. . . 4 |- F/x(y e. B /\ ps)
9	5, 8	eean 1912	. . 3 |- (E.xE.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 240	. 2 |- (E.xE.y((x e. A /\ y e. B) /\ (ph /\ ps)) <-> (E.x(x e. A /\ ph) /\ E.y(y e. B /\ ps)))
11		r2ex 2652	. 2 |- (E.x e. A E.y e. B (ph /\ ps) <-> E.xE.y((x e. A /\ y e. B) /\ (ph /\ ps)))
12		df-rex 2620	. . 3 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
13		df-rex 2620	. . 3 |- (E.y e. B ps <-> E.y(y e. B /\ ps))
14	12, 13	anbi12i 678	. 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 268	1 |- (E.x e. A E.y e. B (ph /\ ps) <-> (E.x e. A ph /\ E.y e. B ps))

Syntax hints:   <-> wb 176   /\ wa 358  E.wex 1541   F/wnf 1544   e. wcel 1710  E.wrex 2615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620

This theorem is referenced by:  reeanv  2778