Theorem rmobidva

Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 16-Jun-2017) Avoid ax-6 , ax-7 , ax-12 . (Revised by Wolf Lammen, 23-Nov-2024)

	Ref	Expression
Hypothesis	rmobidva.1	|- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
Assertion	rmobidva	|- ( ph -> ( E* x e. A ps <-> E* x e. A ch ) )

Proof

Step	Hyp	Ref	Expression
1		rmobidva.1	|- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
2	1	pm5.32da	|- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. A /\ ch ) ) )
3	2	mobidv	|- ( ph -> ( E* x ( x e. A /\ ps ) <-> E* x ( x e. A /\ ch ) ) )
4		df-rmo	|- ( E* x e. A ps <-> E* x ( x e. A /\ ps ) )
5		df-rmo	|- ( E* x e. A ch <-> E* x ( x e. A /\ ch ) )
6	3 4 5	3bitr4g	|- ( ph -> ( E* x e. A ps <-> E* x e. A ch ) )