Theorem cbvexdva 2011

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

Hypothesis

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

Assertion

Ref	Expression
cbvexdva	|- (ph -> (E.xps <-> E.ych))

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

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

Proof of Theorem cbvexdva

Step	Hyp	Ref	Expression
1		nfv 1619	. 2 |- F/yph
2		nfvd 1620	. 2 |- (ph -> F/yps)
3		cbvaldva.1	. . 3 |- ((ph /\ x = y) -> (ps <-> ch))
4	3	ex 423	. 2 |- (ph -> (x = y -> (ps <-> ch)))
5	1, 2, 4	cbvexd 2009	1 |- (ph -> (E.xps <-> E.ych))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  E.wex 1541

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

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  cbvrexdva2  2841