Theorem rmoeqbidv

Description: Formula-building rule for restricted at-most-one quantifier. Deduction form. General version of rmobidv . (Contributed by GG, 1-Sep-2025)

	Ref	Expression
Hypotheses	rmoeqbidv.1	|- ( ph -> A = B )
	rmoeqbidv.2	|- ( ph -> ( ps <-> ch ) )
Assertion	rmoeqbidv	|- ( ph -> ( E* x e. A ps <-> E* x e. B ch ) )

Proof

Step	Hyp	Ref	Expression
1		rmoeqbidv.1	|- ( ph -> A = B )
2		rmoeqbidv.2	|- ( ph -> ( ps <-> ch ) )
3	1	eleq2d	|- ( ph -> ( x e. A <-> x e. B ) )
4	3 2	anbi12d	|- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. B /\ ch ) ) )
5	4	mobidv	|- ( ph -> ( E* x ( x e. A /\ ps ) <-> E* x ( x e. B /\ ch ) ) )
6		df-rmo	|- ( E* x e. A ps <-> E* x ( x e. A /\ ps ) )
7		df-rmo	|- ( E* x e. B ch <-> E* x ( x e. B /\ ch ) )
8	5 6 7	3bitr4g	|- ( ph -> ( E* x e. A ps <-> E* x e. B ch ) )