Theorem rmoeqbii

Description: Equality inference for restricted at-most-one quantifier. (Contributed by GG, 1-Sep-2025)

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

Proof

Step	Hyp	Ref	Expression
1		rmoeqbii.1	|- A = B
2		rmoeqbii.2	|- ( ps <-> ch )
3	1	eleq2i	|- ( x e. A <-> x e. B )
4	3 2	anbi12i	|- ( ( x e. A /\ ps ) <-> ( x e. B /\ ch ) )
5	4	mobii	|- ( 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	3bitr4i	|- ( E* x e. A ps <-> E* x e. B ch )