Theorem rmobiia 2684

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 2079	. 2 |- ( E* x ( x e. A /\ ph ) <-> E* x ( x e. A /\ ps ) )
4		df-rmo 2480	. 2 |- ( E* x e. A ph <-> E* x ( x e. A /\ ph ) )
5		df-rmo 2480	. 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 2043   e. wcel 2164   E*wrmo 2475

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-eu 2045  df-mo 2046  df-rmo 2480

This theorem is referenced by:  rmobii  2685