Theorem cbvrexdva 2777

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 2232	. 2 |- ( ( ph /\ x = y ) -> A = A )
3	1, 2	cbvrexdva2 2775	1 |- ( ph -> ( E. x e. A ps <-> E. y e. A ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   E.wrex 2511

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 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-cleq 2224  df-clel 2227  df-rex 2516

This theorem is referenced by:  tfrlem3ag  6474  tfrlem3a  6475  tfrlemi1  6497  tfr1onlem3ag  6502