Theorem rmoeqi

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

	Ref	Expression
Hypothesis	rmoeqi.1	⊢ 𝐴 = 𝐵
Assertion	rmoeqi	⊢ ( ∃* 𝑥 ∈ 𝐴 𝜓 ↔ ∃* 𝑥 ∈ 𝐵 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		rmoeqi.1	⊢ 𝐴 = 𝐵
2	1	eleq2i	⊢ ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 )
3	2	anbi1i	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝜓 ) )
4	3	mobii	⊢ ( ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ↔ ∃* 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜓 ) )
5		df-rmo	⊢ ( ∃* 𝑥 ∈ 𝐴 𝜓 ↔ ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) )
6		df-rmo	⊢ ( ∃* 𝑥 ∈ 𝐵 𝜓 ↔ ∃* 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜓 ) )
7	4 5 6	3bitr4i	⊢ ( ∃* 𝑥 ∈ 𝐴 𝜓 ↔ ∃* 𝑥 ∈ 𝐵 𝜓 )