Theorem cbvrexdva 2843

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 2354	. 2 |- ((ph /\ x = y) -> A = A)
3	1, 2	cbvrexdva2 2841	1 |- (ph -> (E.x e. A ps <-> E.y e. A ch))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  E.wrex 2616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-cleq 2346  df-clel 2349  df-rex 2621

This theorem is referenced by: (None)