Theorem rmobiia

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

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

Proof

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