Theorem cbvaldva 1844

Description: Rule used to change the bound variable in a universal quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)

Hypothesis

Ref	Expression
cbvaldva.1	|- ( ( ph /\ x = y ) -> ( ps <-> ch ) )

Assertion

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

Distinct variable groups:   ps, y   ch, x   ph, x   ph, y

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

Proof of Theorem cbvaldva

Step	Hyp	Ref	Expression
1		nfv 1461	. 2 |- F/ y ph
2		nfvd 1462	. 2 |- ( ph -> F/ y ps )
3		cbvaldva.1	. . 3 |- ( ( ph /\ x = y ) -> ( ps <-> ch ) )
4	3	ex 113	. 2 |- ( ph -> ( x = y -> ( ps <-> ch ) ) )
5	1, 2, 4	cbvald 1841	1 |- ( ph -> ( A. x ps <-> A. y ch ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1282

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-8 1435  ax-4 1440  ax-17 1459  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:  cbvraldva2  2581