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	⊢ ( 𝜓 ↔ 𝜒 )
Assertion	rmoeqbii	⊢ ( ∃* 𝑥 ∈ 𝐴 𝜓 ↔ ∃* 𝑥 ∈ 𝐵 𝜒 )

Proof

Step	Hyp	Ref	Expression
1		rmoeqbii.1	⊢ 𝐴 = 𝐵
2		rmoeqbii.2	⊢ ( 𝜓 ↔ 𝜒 )
3	1	eleq2i	⊢ ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 )
4	3 2	anbi12i	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝜒 ) )
5	4	mobii	⊢ ( ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ↔ ∃* 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜒 ) )
6		df-rmo	⊢ ( ∃* 𝑥 ∈ 𝐴 𝜓 ↔ ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) )
7		df-rmo	⊢ ( ∃* 𝑥 ∈ 𝐵 𝜒 ↔ ∃* 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜒 ) )
8	5 6 7	3bitr4i	⊢ ( ∃* 𝑥 ∈ 𝐴 𝜓 ↔ ∃* 𝑥 ∈ 𝐵 𝜒 )