Theorem rmobidva 2678

Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 16-Jun-2017.)

Hypothesis

Ref	Expression
rmobidva.1	|- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )

Assertion

Ref	Expression
rmobidva	|- ( ph -> ( E* x e. A ps <-> E* x e. A ch ) )

Distinct variable group:   ph, x

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

Proof of Theorem rmobidva

Step	Hyp	Ref	Expression
1		nfv 1539	. 2 |- F/ x ph
2		rmobidva.1	. 2 |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
3	1, 2	rmobida 2677	1 |- ( ph -> ( E* x e. A ps <-> E* x e. A ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2160   E*wrmo 2471

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-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-eu 2041  df-mo 2042  df-rmo 2476

This theorem is referenced by:  rmobidv  2679