Theorem r2ex 1738

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 groups:   x,y   y,A

Proof of Theorem r2ex

Step	Hyp	Ref	Expression
1		df-rex 1697	. 2 |- (E.x e. A E.y e. B ph <-> E.x(x e. A /\ E.y e. B ph))
2		19.42v 1350	. . . 4 |- (E.y(x e. A /\ (y e. B /\ ph)) <-> (x e. A /\ E.y(y e. B /\ ph)))
3		anass 450	. . . . 5 |- (((x e. A /\ y e. B) /\ ph) <-> (x e. A /\ (y e. B /\ ph)))
4	3	exbii 1092	. . . 4 |- (E.y((x e. A /\ y e. B) /\ ph) <-> E.y(x e. A /\ (y e. B /\ ph)))
5		df-rex 1697	. . . . 5 |- (E.y e. B ph <-> E.y(y e. B /\ ph))
6	5	anbi2i 491	. . . 4 |- ((x e. A /\ E.y e. B ph) <-> (x e. A /\ E.y(y e. B /\ ph)))
7	2, 4, 6	3bitr4i 190	. . 3 |- (E.y((x e. A /\ y e. B) /\ ph) <-> (x e. A /\ E.y e. B ph))
8	7	exbii 1092	. 2 |- (E.xE.y((x e. A /\ y e. B) /\ ph) <-> E.x(x e. A /\ E.y e. B ph))
9	1, 8	bitr4i 183	1 |- (E.x e. A E.y e. B ph <-> E.xE.y((x e. A /\ y e. B) /\ ph))

Syntax hints:   <-> wb 153   /\ wa 230   e. wcel 999  E.wex 1021  E.wrex 1693

This theorem is referenced by:  rexcom 1822  genpv 5167  axcnre 5351  pjtheui 9318  pjpj0i 9338  spanuni 9550  osumlem7 9667  5oalem7 9688  3oalem3 9692  bsi 10589

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1004  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-ex 1022  df-rex 1697

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