Theorem rmobiia 2667

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

Hypothesis

Ref	Expression
rmobiia.1	|- ( x e. A -> ( ph <-> ps ) )

Assertion

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

Proof of Theorem rmobiia

Step	Hyp	Ref	Expression
1		rmobiia.1	. . . 4 |- ( x e. A -> ( ph <-> ps ) )
2	1	pm5.32i 454	. . 3 |- ( ( x e. A /\ ph ) <-> ( x e. A /\ ps ) )
3	2	mobii 2063	. 2 |- ( E* x ( x e. A /\ ph ) <-> E* x ( x e. A /\ ps ) )
4		df-rmo 2463	. 2 |- ( E* x e. A ph <-> E* x ( x e. A /\ ph ) )
5		df-rmo 2463	. 2 |- ( E* x e. A ps <-> E* x ( x e. A /\ ps ) )
6	3, 4, 5	3bitr4i 212	1 |- ( E* x e. A ph <-> E* x e. A ps )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   E*wmo 2027   e. wcel 2148   E*wrmo 2458

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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-eu 2029  df-mo 2030  df-rmo 2463

This theorem is referenced by:  rmobii  2668