Theorem cbv3v 1737

Description: Rule used to change bound variables, using implicit substitution. Version of cbv3 1735 with a disjoint variable condition. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 31-May-2019.)

Hypotheses

Ref	Expression
cbv3v.nf1	|- F/ y ph
cbv3v.nf2	|- F/ x ps
cbv3v.1	|- ( x = y -> ( ph -> ps ) )

Assertion

Ref	Expression
cbv3v	|- ( A. x ph -> A. y ps )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)   ps( x, y)

Proof of Theorem cbv3v

Step	Hyp	Ref	Expression
1		cbv3v.nf1	. 2 |- F/ y ph
2		cbv3v.nf2	. 2 |- F/ x ps
3		cbv3v.1	. 2 |- ( x = y -> ( ph -> ps ) )
4	1, 2, 3	cbv3 1735	1 |- ( A. x ph -> A. y ps )

Syntax hints:   -> wi 4   A.wal 1346   F/wnf 1453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-i9 1523  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-nf 1454

This theorem is referenced by:  cbv1v  1740  cbvalv1  1744