Theorem cbvalv1 2175

Description: Version of cbval 2271 with a dv condition, which does not require ax-13 2246. See cbvalvw 1969 for a version with two dv conditions, requiring fewer axioms, and cbvalv 2273 for another variant. (Contributed by BJ, 31-May-2019.)

Hypotheses

Ref	Expression
cbvalv1.nf1	|- F/ y ph
cbvalv1.nf2	|- F/ x ps
cbvalv1.1	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbvalv1	|- ( A. x ph <-> A. y ps )

Distinct variable group:   x, y

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

Proof of Theorem cbvalv1

Step	Hyp	Ref	Expression
1		cbvalv1.nf1	. . 3 |- F/ y ph
2		cbvalv1.nf2	. . 3 |- F/ x ps
3		cbvalv1.1	. . . 4 |- ( x = y -> ( ph <-> ps ) )
4	3	biimpd 219	. . 3 |- ( x = y -> ( ph -> ps ) )
5	1, 2, 4	cbv3v 2172	. 2 |- ( A. x ph -> A. y ps )
6	3	biimprd 238	. . . 4 |- ( x = y -> ( ps -> ph ) )
7	6	equcoms 1947	. . 3 |- ( y = x -> ( ps -> ph ) )
8	2, 1, 7	cbv3v 2172	. 2 |- ( A. y ps -> A. x ph )
9	5, 8	impbii 199	1 |- ( A. x ph <-> A. y ps )

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-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by:  cbvexv1  2176  cleqh  2724  bj-cbvalvv  32733  bj-cbval2v  32737  bj-abbi  32775