Theorem cbvrmodavw

Description: Change bound variable in the restricted at-most-one quantifier. Deduction form. (Contributed by GG, 14-Aug-2025)

	Ref	Expression
Hypothesis	cbvrmodavw.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) )
Assertion	cbvrmodavw	⊢ ( 𝜑 → ( ∃* 𝑥 ∈ 𝐴 𝜓 ↔ ∃* 𝑦 ∈ 𝐴 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		cbvrmodavw.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) )
2		eleq1w	⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴 ) )
3	2	adantl	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴 ) )
4	3 1	anbi12d	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ↔ ( 𝑦 ∈ 𝐴 ∧ 𝜒 ) ) )
5	4	cbvmodavw	⊢ ( 𝜑 → ( ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ↔ ∃* 𝑦 ( 𝑦 ∈ 𝐴 ∧ 𝜒 ) ) )
6		df-rmo	⊢ ( ∃* 𝑥 ∈ 𝐴 𝜓 ↔ ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) )
7		df-rmo	⊢ ( ∃* 𝑦 ∈ 𝐴 𝜒 ↔ ∃* 𝑦 ( 𝑦 ∈ 𝐴 ∧ 𝜒 ) )
8	5 6 7	3bitr4g	⊢ ( 𝜑 → ( ∃* 𝑥 ∈ 𝐴 𝜓 ↔ ∃* 𝑦 ∈ 𝐴 𝜒 ) )