Theorem cbv2w 1730

Description: Rule used to change bound variables, using implicit substitution. Version of cbv2 1729 with a disjoint variable condition. (Contributed by NM, 5-Aug-1993.) (Revised by Gino Giotto, 10-Jan-2024.)

Hypotheses

Ref	Expression
cbv2w.1	|- F/ x ph
cbv2w.2	|- F/ y ph
cbv2w.3	|- ( ph -> F/ y ps )
cbv2w.4	|- ( ph -> F/ x ch )
cbv2w.5	|- ( ph -> ( x = y -> ( ps <-> ch ) ) )

Assertion

Ref	Expression
cbv2w	|- ( 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 cbv2w

Step	Hyp	Ref	Expression
1		cbv2w.1	. . 3 |- F/ x ph
2		cbv2w.2	. . 3 |- F/ y ph
3		cbv2w.3	. . 3 |- ( ph -> F/ y ps )
4		cbv2w.4	. . 3 |- ( ph -> F/ x ch )
5		cbv2w.5	. . . 4 |- ( ph -> ( x = y -> ( ps <-> ch ) ) )
6		biimp 117	. . . 4 |- ( ( ps <-> ch ) -> ( ps -> ch ) )
7	5, 6	syl6 33	. . 3 |- ( ph -> ( x = y -> ( ps -> ch ) ) )
8	1, 2, 3, 4, 7	cbv1v 1727	. 2 |- ( ph -> ( A. x ps -> A. y ch ) )
9		equcomi 1684	. . . 4 |- ( y = x -> x = y )
10		biimpr 129	. . . 4 |- ( ( ps <-> ch ) -> ( ch -> ps ) )
11	9, 5, 10	syl56 34	. . 3 |- ( ph -> ( y = x -> ( ch -> ps ) ) )
12	2, 1, 4, 3, 11	cbv1v 1727	. 2 |- ( ph -> ( A. y ch -> A. x ps ) )
13	8, 12	impbid 128	1 |- ( ph -> ( A. x ps <-> A. y ch ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1333   F/wnf 1440

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-nf 1441

This theorem is referenced by: (None)