Theorem cbvreuv

Description: Change the bound variable of a restricted unique existential quantifier using implicit substitution. See cbvreuvw for a version without ax-13 , but extra disjoint variables. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbvreuvw when possible. (Contributed by NM, 5-Apr-2004) (Revised by Mario Carneiro, 15-Oct-2016) (New usage is discouraged.)

	Ref	Expression
Hypothesis	cbvrmov.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	cbvreuv	$$⊢\exists !x\in A\phi \leftrightarrow \exists !y\in A\psi$$

Proof

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