Theorem cbvexdva 1804

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 1421	. 2 |- F/ y ph
2		nfvd 1422	. 2 |- ( ph -> F/ y ps )
3		cbvaldva.1	. . 3 |- ( ( ph /\ x = y ) -> ( ps <-> ch ) )
4	3	ex 108	. 2 |- ( ph -> ( x = y -> ( ps <-> ch ) ) )
5	1, 2, 4	cbvexd 1802	1 |- ( ph -> ( E. x ps <-> E. y ch ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   E.wex 1381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-nf 1350

This theorem is referenced by:  cbvrexdva2  2538  acexmid  5511  tfrlemi1  5946  ltexpri  6711  recexpr  6736