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	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	cbvreuvw	$$⊢\exists !x\in A\phi \leftrightarrow \exists !y\in A\psi$$

Proof

Step	Hyp	Ref	Expression
1		cbvralvw.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
2		nfv	$$⊢Ⅎy\phi$$
3		nfv	$$⊢Ⅎx\psi$$
4	2 3 1	cbvreuw	$$⊢\exists !x\in A\phi \leftrightarrow \exists !y\in A\psi$$