Theorem cbvald 1800

Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 1893. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Revised by Wolf Lammen, 13-May-2018.)

Hypotheses

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

Assertion

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

Distinct variable groups:   ph, x   ch, x

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

Proof of Theorem cbvald

Step	Hyp	Ref	Expression
1		nfv 1421	. 2 |- F/ x ph
2		cbvald.1	. 2 |- F/ y ph
3		cbvald.2	. 2 |- ( ph -> F/ y ps )
4		nfv 1421	. . 3 |- F/ x ch
5	4	a1i 9	. 2 |- ( ph -> F/ x ch )
6		cbvald.3	. 2 |- ( ph -> ( x = y -> ( ps <-> ch ) ) )
7	1, 2, 3, 5, 6	cbv2 1635	1 |- ( ph -> ( A. x ps <-> A. y ch ) )

Syntax hints:   -> wi 4   <-> wb 98   A.wal 1241   F/wnf 1349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350

This theorem is referenced by:  cbvaldva  1803