Theorem cbvalv1 1761

Description: Rule used to change bound variables, using implicit substitution. Version of cbval 1764 with a disjoint variable condition. See cbvalvw 1929 for a version with two disjoint variable conditions, and cbvalv 1927 for another variant. (Contributed by NM, 13-May-1993.) (Revised 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 144	. . 3 |- ( x = y -> ( ph -> ps ) )
5	1, 2, 4	cbv3v 1754	. 2 |- ( A. x ph -> A. y ps )
6	3	biimprd 158	. . . 4 |- ( x = y -> ( ps -> ph ) )
7	6	equcoms 1718	. . 3 |- ( y = x -> ( ps -> ph ) )
8	2, 1, 7	cbv3v 1754	. 2 |- ( A. y ps -> A. x ph )
9	5, 8	impbii 126	1 |- ( A. x ph <-> A. y ps )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1361   F/wnf 1470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544

This theorem depends on definitions:  df-bi 117  df-nf 1471

This theorem is referenced by:  cbvralfw  2705