Theorem cbvexdva 1899

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. x ps <-> E. y ch ) )

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 1508	. 2 |- F/ y ph
2		nfvd 1509	. 2 |- ( ph -> F/ y ps )
3		cbvaldva.1	. . 3 |- ( ( ph /\ x = y ) -> ( ps <-> ch ) )
4	3	ex 114	. 2 |- ( ph -> ( x = y -> ( ps <-> ch ) ) )
5	1, 2, 4	cbvexd 1897	1 |- ( ph -> ( E. x ps <-> E. y ch ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   E.wex 1468

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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-nf 1437

This theorem is referenced by:  cbvrexdva2  2657  acexmid  5766  tfrlemi1  6222  ltexpri  7414  recexpr  7439