Theorem rmobidv 2542

Description: Formula-building rule for restricted existential quantifier (deduction rule). (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 270	. 2 |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
3	2	rmobidva 2541	1 |- ( ph -> ( E* x e. A ps <-> E* x e. A ch ) )

Syntax hints:   -> wi 4   <-> wb 103   e. wcel 1433   E*wrmo 2351

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-eu 1944  df-mo 1945  df-rmo 2356

This theorem is referenced by:  rmoeqd  2560