Theorem bj-cbv2v 32732

Description: Version of cbv2 2270 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-cbv2v.1	|- F/ x ph
bj-cbv2v.2	|- F/ y ph
bj-cbv2v.3	|- ( ph -> F/ y ps )
bj-cbv2v.4	|- ( ph -> F/ x ch )
bj-cbv2v.5	|- ( ph -> ( x = y -> ( ps <-> ch ) ) )

Assertion

Ref	Expression
bj-cbv2v	|- ( ph -> ( A. x ps <-> A. y ch ) )

Distinct variable group:   x, y

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

Proof of Theorem bj-cbv2v

Step	Hyp	Ref	Expression
1		bj-cbv2v.2	. . . 4 |- F/ y ph
2	1	nf5ri 2065	. . 3 |- ( ph -> A. y ph )
3		bj-cbv2v.1	. . . . 5 |- F/ x ph
4	3	nfal 2153	. . . 4 |- F/ x A. y ph
5	4	nf5ri 2065	. . 3 |- ( A. y ph -> A. x A. y ph )
6	2, 5	syl 17	. 2 |- ( ph -> A. x A. y ph )
7		bj-cbv2v.3	. . . 4 |- ( ph -> F/ y ps )
8	7	nf5rd 2066	. . 3 |- ( ph -> ( ps -> A. y ps ) )
9		bj-cbv2v.4	. . . 4 |- ( ph -> F/ x ch )
10	9	nf5rd 2066	. . 3 |- ( ph -> ( ch -> A. x ch ) )
11		bj-cbv2v.5	. . 3 |- ( ph -> ( x = y -> ( ps <-> ch ) ) )
12	8, 10, 11	bj-cbv2hv 32731	. 2 |- ( A. x A. y ph -> ( A. x ps <-> A. y ch ) )
13	6, 12	syl 17	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-cbvaldv  32735