Theorem cbvrexdva 2584

Description: Rule used to change the bound variable in a restricted existential 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
cbvrexdva	|- ( ph -> ( E. x e. A ps <-> E. 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 cbvrexdva

Step	Hyp	Ref	Expression
1		cbvraldva.1	. 2 |- ( ( ph /\ x = y ) -> ( ps <-> ch ) )
2		eqidd 2082	. 2 |- ( ( ph /\ x = y ) -> A = A )
3	1, 2	cbvrexdva2 2582	1 |- ( ph -> ( E. x e. A ps <-> E. y e. A ch ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   E.wrex 2349

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-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-cleq 2074  df-clel 2077  df-rex 2354

This theorem is referenced by:  tfrlem3ag  5947  tfrlem3a  5948  tfrlemi1  5969