Theorem r2ex 1688

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 1647	. 2 |- (E.x e. A E.y e. B ph <-> E.x(x e. A /\ E.y e. B ph))
2		19.42v 1306	. . . 4 |- (E.y(x e. A /\ (y e. B /\ ph)) <-> (x e. A /\ E.y(y e. B /\ ph)))
3		anass 439	. . . . 5 |- (((x e. A /\ y e. B) /\ ph) <-> (x e. A /\ (y e. B /\ ph)))
4	3	exbii 1049	. . . 4 |- (E.y((x e. A /\ y e. B) /\ ph) <-> E.y(x e. A /\ (y e. B /\ ph)))
5		df-rex 1647	. . . . 5 |- (E.y e. B ph <-> E.y(y e. B /\ ph))
6	5	anbi2i 480	. . . 4 |- ((x e. A /\ E.y e. B ph) <-> (x e. A /\ E.y(y e. B /\ ph)))
7	2, 4, 6	3bitr4 183	. . 3 |- (E.y((x e. A /\ y e. B) /\ ph) <-> (x e. A /\ E.y e. B ph))
8	7	exbii 1049	. 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 176	1 |- (E.x e. A E.y e. B ph <-> E.xE.y((x e. A /\ y e. B) /\ ph))

Syntax hints:   <-> wb 146   /\ wa 223   e. wcel 956  E.wex 978  E.wrex 1643

This theorem is referenced by:  rexcom 1772  genpv 5082  axcnre 5266  pjtheu 9173  pjpj0 9193  spanun 9405  osumlem7 9524  5oalem7 9545  3oalem3 9549  bsi 10418

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 961  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-rex 1647

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