Theorem cbvreuvw

Description: Change the bound variable of a restricted unique existential quantifier using implicit substitution. Version of cbvreuv with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 5-Apr-2004) (Revised by Gino Giotto, 10-Jan-2024)

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

Proof

Step	Hyp	Ref	Expression
1		cbvralvw.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
2		nfv	⊢ Ⅎ 𝑦 𝜑
3		nfv	⊢ Ⅎ 𝑥 𝜓
4	2 3 1	cbvreuw	⊢ ( ∃! 𝑥 ∈ 𝐴 𝜑 ↔ ∃! 𝑦 ∈ 𝐴 𝜓 )