Theorem cbvexdvaw 1943

Description: Rule used to change the bound variable in an existential quantifier with implicit substitution. Deduction form. Version of cbvexdva 1941 with a disjoint variable condition. (Contributed by David Moews, 1-May-2017.) (Revised by GG, 10-Jan-2024.) (Revised by Wolf Lammen, 10-Feb-2024.)

Hypothesis

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

Assertion

Ref	Expression
cbvexdvaw	|- ( ph -> ( E. x ps <-> E. y ch ) )

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

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

Proof of Theorem cbvexdvaw

Step	Hyp	Ref	Expression
1		cbvaldvaw.1	. 2 |- ( ( ph /\ x = y ) -> ( ps <-> ch ) )
2	1	cbvexdva 1941	1 |- ( ph -> ( E. x ps <-> E. y ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   E.wex 1503

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-nf 1472

This theorem is referenced by:  cbvex2vw  1945