Theorem cbvexdva 1953

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 1551	. 2 |- F/ y ph
2		nfvd 1552	. 2 |- ( ph -> F/ y ps )
3		cbvaldva.1	. . 3 |- ( ( ph /\ x = y ) -> ( ps <-> ch ) )
4	3	ex 115	. 2 |- ( ph -> ( x = y -> ( ps <-> ch ) ) )
5	1, 2, 4	cbvexd 1951	1 |- ( ph -> ( E. x ps <-> E. y ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   E.wex 1515

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557

This theorem depends on definitions:  df-bi 117  df-nf 1484

This theorem is referenced by:  cbvexdvaw  1955  cbvrexdva2  2746  acexmid  5945  tfrlemi1  6420  ltexpri  7728  recexpr  7753