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	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	rmoeqbidv.2	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
Assertion	rmoeqbidv	⊢ ( 𝜑 → ( ∃* 𝑥 ∈ 𝐴 𝜓 ↔ ∃* 𝑥 ∈ 𝐵 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		rmoeqbidv.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		rmoeqbidv.2	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
3	1	eleq2d	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) )
4	3 2	anbi12d	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝜒 ) ) )
5	4	mobidv	⊢ ( 𝜑 → ( ∃* 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	3bitr4g	⊢ ( 𝜑 → ( ∃* 𝑥 ∈ 𝐴 𝜓 ↔ ∃* 𝑥 ∈ 𝐵 𝜒 ) )