Theorem cbvrexdva 2702

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 2166	. 2 |- ( ( ph /\ x = y ) -> A = A )
3	1, 2	cbvrexdva2 2700	1 |- ( ph -> ( E. x e. A ps <-> E. y e. A ch ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   E.wrex 2445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-cleq 2158  df-clel 2161  df-rex 2450

This theorem is referenced by:  tfrlem3ag  6277  tfrlem3a  6278  tfrlemi1  6300  tfr1onlem3ag  6305