Theorem bj-cbvaldv 32735

Description: Version of cbvald 2277 with a dv condition, which does not require ax-13 2246. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-cbvaldv.1	|- F/ y ph
bj-cbvaldv.2	|- ( ph -> F/ y ps )
bj-cbvaldv.3	|- ( ph -> ( x = y -> ( ps <-> ch ) ) )

Assertion

Ref	Expression
bj-cbvaldv	|- ( ph -> ( A. x ps <-> A. y ch ) )

Distinct variable groups:   x, y   ph, x   ch, x

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

Proof of Theorem bj-cbvaldv

Step	Hyp	Ref	Expression
1		nfv 1843	. 2 |- F/ x ph
2		bj-cbvaldv.1	. 2 |- F/ y ph
3		bj-cbvaldv.2	. 2 |- ( ph -> F/ y ps )
4		nfv 1843	. . 3 |- F/ x ch
5	4	a1i 11	. 2 |- ( ph -> F/ x ch )
6		bj-cbvaldv.3	. 2 |- ( ph -> ( x = y -> ( ps <-> ch ) ) )
7	1, 2, 3, 5, 6	bj-cbv2v 32732	1 |- ( ph -> ( A. x ps <-> A. y ch ) )

Syntax hints:   -> wi 4   <-> wb 196   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-or 385  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by:  bj-cbvexdv  32736  bj-cbvaldvav  32741