Theorem cbv1v 1727

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 16-Jun-2019.)

Hypotheses

Ref	Expression
cbv1v.1	|- F/ x ph
cbv1v.2	|- F/ y ph
cbv1v.3	|- ( ph -> F/ y ps )
cbv1v.4	|- ( ph -> F/ x ch )
cbv1v.5	|- ( ph -> ( x = y -> ( ps -> ch ) ) )

Assertion

Ref	Expression
cbv1v	|- ( 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 cbv1v

Step	Hyp	Ref	Expression
1		cbv1v.2	. . . . 5 |- F/ y ph
2		cbv1v.3	. . . . 5 |- ( ph -> F/ y ps )
3	1, 2	nfim1 1551	. . . 4 |- F/ y ( ph -> ps )
4		cbv1v.1	. . . . 5 |- F/ x ph
5		cbv1v.4	. . . . 5 |- ( ph -> F/ x ch )
6	4, 5	nfim1 1551	. . . 4 |- F/ x ( ph -> ch )
7		cbv1v.5	. . . . . 6 |- ( ph -> ( x = y -> ( ps -> ch ) ) )
8	7	com12 30	. . . . 5 |- ( x = y -> ( ph -> ( ps -> ch ) ) )
9	8	a2d 26	. . . 4 |- ( x = y -> ( ( ph -> ps ) -> ( ph -> ch ) ) )
10	3, 6, 9	cbv3v 1724	. . 3 |- ( A. x ( ph -> ps ) -> A. y ( ph -> ch ) )
11	4	19.21 1563	. . 3 |- ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) )
12	1	19.21 1563	. . 3 |- ( A. y ( ph -> ch ) <-> ( ph -> A. y ch ) )
13	10, 11, 12	3imtr3i 199	. 2 |- ( ( ph -> A. x ps ) -> ( ph -> A. y ch ) )
14	13	pm2.86i 98	1 |- ( ph -> ( A. x ps -> A. y ch ) )

Syntax hints:   -> wi 4   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-4 1490  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:  cbv2w  1730