Theorem reeanv 1316

Description: Rearrange existential quantifiers.

Assertion

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

Distinct variable group(s):   ph,y   ps,x   x,y   y,A   x,B

Proof of Theorem reeanv

Step	Hyp	Ref	Expression
1		anass 336	. . . . 5 |- (((x e. A /\ y e. B) /\ (ph /\ ps)) <-> (x e. A /\ (y e. B /\ (ph /\ ps))))
2		an4 388	. . . . 5 |- (((x e. A /\ y e. B) /\ (ph /\ ps)) <-> ((x e. A /\ ph) /\ (y e. B /\ ps)))
3	1, 2	bitr3 153	. . . 4 |- ((x e. A /\ (y e. B /\ (ph /\ ps))) <-> ((x e. A /\ ph) /\ (y e. B /\ ps)))
4	3	bi2ex 734	. . 3 |- (E.xE.y(x e. A /\ (y e. B /\ (ph /\ ps))) <-> E.xE.y((x e. A /\ ph) /\ (y e. B /\ ps)))
5		exdistr 967	. . 3 |- (E.xE.y(x e. A /\ (y e. B /\ (ph /\ ps))) <-> E.x(x e. A /\ E.y(y e. B /\ (ph /\ ps))))
6		eeanv 980	. . 3 |- (E.xE.y((x e. A /\ ph) /\ (y e. B /\ ps)) <-> (E.x(x e. A /\ ph) /\ E.y(y e. B /\ ps)))
7	4, 5, 6	3bitr3 156	. 2 |- (E.x(x e. A /\ E.y(y e. B /\ (ph /\ ps))) <-> (E.x(x e. A /\ ph) /\ E.y(y e. B /\ ps)))
8		df-rex 1206	. . . 4 |- (E.y e. B (ph /\ ps) <-> E.y(y e. B /\ (ph /\ ps)))
9	8	birex 1224	. . 3 |- (E.x e. A E.y e. B (ph /\ ps) <-> E.x e. A E.y(y e. B /\ (ph /\ ps)))
10		df-rex 1206	. . 3 |- (E.x e. A E.y(y e. B /\ (ph /\ ps)) <-> E.x(x e. A /\ E.y(y e. B /\ (ph /\ ps))))
11	9, 10	bitr 151	. 2 |- (E.x e. A E.y e. B (ph /\ ps) <-> E.x(x e. A /\ E.y(y e. B /\ (ph /\ ps))))
12		df-rex 1206	. . 3 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
13		df-rex 1206	. . 3 |- (E.y e. B ps <-> E.y(y e. B /\ ps))
14	12, 13	anbi12i 369	. 2 |- ((E.x e. A ph /\ E.y e. B ps) <-> (E.x(x e. A /\ ph) /\ E.y(y e. B /\ ps)))
15	7, 11, 14	3bitr4 158	1 |- (E.x e. A E.y e. B (ph /\ ps) <-> (E.x e. A ph /\ E.y e. B ps))

Syntax hints:   <-> wb 127   /\ wa 196  E.wex 678   e. wcel 1092  E.wrex 1202

This theorem is referenced by:  tfrlem5 2953  unfi 3441  kmlem8 3587  climunii 4883  infxpidmlem7 4939  hlimunii 5143  pjthu 5241  pjthu2 5242  pjpj0 5259

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-17 925

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-rex 1206

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