Theorem cbvraldva 2795

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

Hypothesis

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

Assertion

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

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

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

Proof of Theorem cbvraldva

Step	Hyp	Ref	Expression
1		cbvraldva.1	. 2 |- ( ( ph /\ x = y ) -> ( ps <-> ch ) )
2		eqidd 2239	. 2 |- ( ( ph /\ x = y ) -> A = A )
3	1, 2	cbvraldva2 2793	1 |- ( ph -> ( A. x e. A ps <-> A. y e. A ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wral 2528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-cleq 2231  df-clel 2234  df-ral 2533

This theorem is referenced by:  wrd2ind  11476