Theorem cbvmodavw

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

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

Proof

Step	Hyp	Ref	Expression
1		cbvmodavw.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) )
2		equequ1	⊢ ( 𝑥 = 𝑦 → ( 𝑥 = 𝑧 ↔ 𝑦 = 𝑧 ) )
3	2	adantl	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝑥 = 𝑧 ↔ 𝑦 = 𝑧 ) )
4	1 3	imbi12d	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( ( 𝜓 → 𝑥 = 𝑧 ) ↔ ( 𝜒 → 𝑦 = 𝑧 ) ) )
5	4	cbvaldvaw	⊢ ( 𝜑 → ( ∀ 𝑥 ( 𝜓 → 𝑥 = 𝑧 ) ↔ ∀ 𝑦 ( 𝜒 → 𝑦 = 𝑧 ) ) )
6	5	exbidv	⊢ ( 𝜑 → ( ∃ 𝑧 ∀ 𝑥 ( 𝜓 → 𝑥 = 𝑧 ) ↔ ∃ 𝑧 ∀ 𝑦 ( 𝜒 → 𝑦 = 𝑧 ) ) )
7		dfmo	⊢ ( ∃* 𝑥 𝜓 ↔ ∃ 𝑧 ∀ 𝑥 ( 𝜓 → 𝑥 = 𝑧 ) )
8		dfmo	⊢ ( ∃* 𝑦 𝜒 ↔ ∃ 𝑧 ∀ 𝑦 ( 𝜒 → 𝑦 = 𝑧 ) )
9	6 7 8	3bitr4g	⊢ ( 𝜑 → ( ∃* 𝑥 𝜓 ↔ ∃* 𝑦 𝜒 ) )