Theorem cbv3v 2172

Description: Version of cbv3 2265 with a dv condition, which does not require ax-13 2246. (Contributed 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	. . 3 |- F/ y ph
2	1	nfal 2153	. 2 |- F/ y A. x ph
3		cbv3v.nf2	. . 3 |- F/ x ps
4		cbv3v.1	. . 3 |- ( x = y -> ( ph -> ps ) )
5	3, 4	spimv1 2115	. 2 |- ( A. x ph -> ps )
6	2, 5	alrimi 2082	1 |- ( A. x ph -> A. y ps )

Syntax hints:   -> wi 4   A.wal 1481   F/wnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-ex 1705  df-nf 1710

This theorem is referenced by:  cbv3hv  2174  cbvalv1  2175  bj-cbv1v  32729