Theorem ralcom 1312

Description: Commutation of restricted quantifiers.

Assertion

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

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

Proof of Theorem ralcom

Step	Hyp	Ref	Expression
1		ancom 333	. . . . 5 |- ((x e. A /\ y e. B) <-> (y e. B /\ x e. A))
2	1	imbi1i 161	. . . 4 |- (((x e. A /\ y e. B) -> ph) <-> ((y e. B /\ x e. A) -> ph))
3	2	bi2al 696	. . 3 |- (A.xA.y((x e. A /\ y e. B) -> ph) <-> A.xA.y((y e. B /\ x e. A) -> ph))
4		alcom 715	. . 3 |- (A.xA.y((y e. B /\ x e. A) -> ph) <-> A.yA.x((y e. B /\ x e. A) -> ph))
5	3, 4	bitr 151	. 2 |- (A.xA.y((x e. A /\ y e. B) -> ph) <-> A.yA.x((y e. B /\ x e. A) -> ph))
6		r2al 1231	. 2 |- (A.x e. A A.y e. B ph <-> A.xA.y((x e. A /\ y e. B) -> ph))
7		r2al 1231	. 2 |- (A.y e. B A.x e. A ph <-> A.yA.x((y e. B /\ x e. A) -> ph))
8	5, 6, 7	3bitr4 158	1 |- (A.x e. A A.y e. B ph <-> A.y e. B A.x e. A ph)

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672   e. wcel 1092  A.wral 1201

This theorem is referenced by:  ralcom4 1360  ssint 1980  fununi 2705  mapxpen 3390  occl 5188

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-an 198  df-ral 1205

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