Theorem r2ex 1241

Description: Double restricted existential quantification.

Assertion

Ref	Expression
r2ex	|- (E.x e. A E.y e. B ph <-> E.xE.y((x e. A /\ y e. B) /\ ph))

Distinct variable group(s):   x,y   y,A

Proof of Theorem r2ex

Step	Hyp	Ref	Expression
1		df-rex 1206	. 2 |- (E.x e. A E.y e. B ph <-> E.x(x e. A /\ E.y e. B ph))
2		19.42v 966	. . . 4 |- (E.y(x e. A /\ (y e. B /\ ph)) <-> (x e. A /\ E.y(y e. B /\ ph)))
3		anass 336	. . . . 5 |- (((x e. A /\ y e. B) /\ ph) <-> (x e. A /\ (y e. B /\ ph)))
4	3	biex 733	. . . 4 |- (E.y((x e. A /\ y e. B) /\ ph) <-> E.y(x e. A /\ (y e. B /\ ph)))
5		df-rex 1206	. . . . 5 |- (E.y e. B ph <-> E.y(y e. B /\ ph))
6	5	anbi2i 367	. . . 4 |- ((x e. A /\ E.y e. B ph) <-> (x e. A /\ E.y(y e. B /\ ph)))
7	2, 4, 6	3bitr4 158	. . 3 |- (E.y((x e. A /\ y e. B) /\ ph) <-> (x e. A /\ E.y e. B ph))
8	7	biex 733	. 2 |- (E.xE.y((x e. A /\ y e. B) /\ ph) <-> E.x(x e. A /\ E.y e. B ph))
9	1, 8	bitr4 154	1 |- (E.x e. A E.y e. B ph <-> E.xE.y((x e. A /\ y e. B) /\ ph))

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

This theorem is referenced by:  rexcom 1313  cbvop 2473  genpv 3896  axcnre 4087  pjthu 5241  pjthu2 5242  pjpj0 5259  spanun 5450  osumlem7 5536  5oalem7 5550  3oalem3 5554

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-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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