Theorem cbvral 1331

Description: Rule used to change bound variables with implicit substitution.

Hypotheses

Ref	Expression
cbvral.1	|- (ph -> A.yph)
cbvral.2	|- (ps -> A.xps)
cbvral.3	|- (x = y -> (ph <-> ps))

Assertion

Ref	Expression
cbvral	|- (A.x e. A ph <-> A.y e. A ps)

Distinct variable group(s):   x,y,A

Proof of Theorem cbvral

Step	Hyp	Ref	Expression
1		ax-17 925	. 2 |- (z e. A -> A.x z e. A)
2		ax-17 925	. 2 |- (z e. A -> A.y z e. A)
3		cbvral.1	. 2 |- (ph -> A.yph)
4		cbvral.2	. 2 |- (ps -> A.xps)
5		cbvral.3	. 2 |- (x = y -> (ph <-> ps))
6	1, 2, 3, 4, 5	cbvralf 1330	1 |- (A.x e. A ph <-> A.y e. A ps)

Syntax hints:   -> wi 2   <-> wb 127  A.wal 672   = weq 797   e. wcel 1092  A.wral 1201

This theorem is referenced by:  cbvralv 1333  sbralie 1439  tfinds 2401  tfindes 2404  cleqfvf 2881  f1fvf 2917  tfrlem1 2949

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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