Theorem rmobida 3129

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

Hypotheses

Ref	Expression
rmobida.1	|- F/ x ph
rmobida.2	|- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )

Assertion

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

Proof of Theorem rmobida

Step	Hyp	Ref	Expression
1		rmobida.1	. . 3 |- F/ x ph
2		rmobida.2	. . . 4 |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
3	2	pm5.32da 673	. . 3 |- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. A /\ ch ) ) )
4	1, 3	mobid 2489	. 2 |- ( ph -> ( E* x ( x e. A /\ ps ) <-> E* x ( x e. A /\ ch ) ) )
5		df-rmo 2920	. 2 |- ( E* x e. A ps <-> E* x ( x e. A /\ ps ) )
6		df-rmo 2920	. 2 |- ( E* x e. A ch <-> E* x ( x e. A /\ ch ) )
7	4, 5, 6	3bitr4g 303	1 |- ( ph -> ( E* x e. A ps <-> E* x e. A ch ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   F/wnf 1708   e. wcel 1990   E*wmo 2471   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:  rmobidva  3130  reuan  41180