Theorem rexcom 1775

Description: Commutation of restricted quantifiers.

Assertion

Ref	Expression
rexcom	|- (E.x e. A E.y e. B ph <-> E.y e. B E.x e. A ph)

Distinct variable groups:   x,y   x,B   y,A

Proof of Theorem rexcom

Step	Hyp	Ref	Expression
1		ancom 435	. . . . 5 |- ((x e. A /\ y e. B) <-> (y e. B /\ x e. A))
2	1	anbi1i 481	. . . 4 |- (((x e. A /\ y e. B) /\ ph) <-> ((y e. B /\ x e. A) /\ ph))
3	2	2exbii 1052	. . 3 |- (E.xE.y((x e. A /\ y e. B) /\ ph) <-> E.xE.y((y e. B /\ x e. A) /\ ph))
4		excom 1046	. . 3 |- (E.xE.y((y e. B /\ x e. A) /\ ph) <-> E.yE.x((y e. B /\ x e. A) /\ ph))
5	3, 4	bitr 173	. 2 |- (E.xE.y((x e. A /\ y e. B) /\ ph) <-> E.yE.x((y e. B /\ x e. A) /\ ph))
6		r2ex 1691	. 2 |- (E.x e. A E.y e. B ph <-> E.xE.y((x e. A /\ y e. B) /\ ph))
7		r2ex 1691	. 2 |- (E.y e. B E.x e. A ph <-> E.yE.x((y e. B /\ x e. A) /\ ph))
8	5, 6, 7	3bitr4 183	1 |- (E.x e. A E.y e. B ph <-> E.y e. B E.x e. A ph)

Syntax hints:   <-> wb 146   /\ wa 223   e. wcel 958  E.wex 980  E.wrex 1646

This theorem is referenced by:  rexcom4 1824  brdom7disj 4804  creui 6743  shscomt 9283  mdsymlem4 10333  mdsymlem8 10337

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-rex 1650

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