Theorem cbv1 1672

Description: Rule used to change bound variables, using implicit substitution. Revised to format hypotheses to common style. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Revised by Wolf Lammen, 13-May-2018.)

Hypotheses

Ref	Expression
cbv1.1	|- F/ x ph
cbv1.2	|- F/ y ph
cbv1.3	|- ( ph -> F/ y ps )
cbv1.4	|- ( ph -> F/ x ch )
cbv1.5	|- ( ph -> ( x = y -> ( ps -> ch ) ) )

Assertion

Ref	Expression
cbv1	|- ( ph -> ( A. x ps -> A. y ch ) )

Proof of Theorem cbv1

Step	Hyp	Ref	Expression
1		cbv1.2	. . . . 5 |- F/ y ph
2		cbv1.3	. . . . 5 |- ( ph -> F/ y ps )
3	1, 2	nfim1 1503	. . . 4 |- F/ y ( ph -> ps )
4		cbv1.1	. . . . 5 |- F/ x ph
5		cbv1.4	. . . . 5 |- ( ph -> F/ x ch )
6	4, 5	nfim1 1503	. . . 4 |- F/ x ( ph -> ch )
7		cbv1.5	. . . . . 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	cbv3 1670	. . 3 |- ( A. x ( ph -> ps ) -> A. y ( ph -> ch ) )
11	4	19.21 1515	. . 3 |- ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) )
12	1	19.21 1515	. . 3 |- ( A. y ( ph -> ch ) <-> ( ph -> A. y ch ) )
13	10, 11, 12	3imtr3i 198	. 2 |- ( ( ph -> A. x ps ) -> ( ph -> A. y ch ) )
14	13	pm2.86i 97	1 |- ( ph -> ( A. x ps -> A. y ch ) )

Syntax hints:   -> wi 4   A.wal 1282   F/wnf 1389

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390

This theorem is referenced by:  cbv1h  1673