Theorem cbvreudavw

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

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

Proof

Step	Hyp	Ref	Expression
1		cbvreudavw.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	cbveudavw	⊢ ( 𝜑 → ( ∃! 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ↔ ∃! 𝑦 ( 𝑦 ∈ 𝐴 ∧ 𝜒 ) ) )
6		df-reu	⊢ ( ∃! 𝑥 ∈ 𝐴 𝜓 ↔ ∃! 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) )
7		df-reu	⊢ ( ∃! 𝑦 ∈ 𝐴 𝜒 ↔ ∃! 𝑦 ( 𝑦 ∈ 𝐴 ∧ 𝜒 ) )
8	5 6 7	3bitr4g	⊢ ( 𝜑 → ( ∃! 𝑥 ∈ 𝐴 𝜓 ↔ ∃! 𝑦 ∈ 𝐴 𝜒 ) )