Theorem cbvral2v 1336

Description: Change bound variables of double restricted universal quantification, using implicit substitution.

Hypotheses

Ref	Expression
cbvral2v.1	|- (x = z -> (ph <-> ch))
cbvral2v.2	|- (y = w -> (ch <-> ps))

Assertion

Ref	Expression
cbvral2v	|- (A.x e. A A.y e. B ph <-> A.z e. A A.w e. B ps)

Distinct variable group(s):   x,z,A   x,y,w,B,z   ph,z   ps,x,y   ch,x,w

Proof of Theorem cbvral2v

Step	Hyp	Ref	Expression
1		cbvral2v.1	. . . 4 |- (x = z -> (ph <-> ch))
2	1	biraldv 1219	. . 3 |- (x = z -> (A.y e. B ph <-> A.y e. B ch))
3	2	cbvralv 1333	. 2 |- (A.x e. A A.y e. B ph <-> A.z e. A A.y e. B ch)
4		cbvral2v.2	. . . 4 |- (y = w -> (ch <-> ps))
5	4	cbvralv 1333	. . 3 |- (A.y e. B ch <-> A.w e. B ps)
6	5	biral 1223	. 2 |- (A.z e. A A.y e. B ch <-> A.z e. A A.w e. B ps)
7	3, 6	bitr 151	1 |- (A.x e. A A.y e. B ph <-> A.z e. A A.w e. B ps)

Syntax hints:   -> wi 2   <-> wb 127   = weq 797  A.wral 1201

This theorem is referenced by:  fununi 2705

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-8 798  ax-9 799  ax-12 802  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-cleq 1097  df-clel 1099  df-ral 1205

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