Theorem rmobidv 2645

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

Hypothesis

Ref	Expression
rmobidv.1	|- ( ph -> ( ps <-> ch ) )

Assertion

Ref	Expression
rmobidv	|- ( 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 rmobidv

Step	Hyp	Ref	Expression
1		rmobidv.1	. . 3 |- ( ph -> ( ps <-> ch ) )
2	1	adantr 274	. 2 |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
3	2	rmobidva 2644	1 |- ( ph -> ( E* x e. A ps <-> E* x e. A ch ) )

Syntax hints:   -> wi 4   <-> wb 104   e. wcel 2128   E*wrmo 2438

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 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-17 1506  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-eu 2009  df-mo 2010  df-rmo 2443

This theorem is referenced by:  rmoeqd  2663  disjxp1  6185