Theorem rmobidv 3131

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 481	. 2 |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
3	2	rmobidva 3130	1 |- ( ph -> ( E* x e. A ps <-> E* x e. A ch ) )

Syntax hints:   -> wi 4   <-> wb 196   e. wcel 1990   E*wrmo 2915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-nf 1710  df-eu 2474  df-mo 2475  df-rmo 2920

This theorem is referenced by:  rmoeqd  3149  brdom7disj  9353  ddemeas  30299  poimirlem26  33435