Theorem cbvrexw

Description: Rule used to change bound variables, using implicit substitution. Version of cbvrex with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 31-Jul-2003) (Revised by Gino Giotto, 10-Jan-2024)

	Ref	Expression
Hypotheses	cbvralw.1	$$⊢Ⅎy\phi$$
	cbvralw.2	$$⊢Ⅎx\psi$$
	cbvralw.3	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	cbvrexw	$$⊢\exists x\in A\phi \leftrightarrow \exists y\in A\psi$$

Proof

Step	Hyp	Ref	Expression
1		cbvralw.1	$$⊢Ⅎy\phi$$
2		cbvralw.2	$$⊢Ⅎx\psi$$
3		cbvralw.3	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
4		nfcv	$$⊢\underset{\_}{Ⅎ}xA$$
5		nfcv	$$⊢\underset{\_}{Ⅎ}yA$$
6	4 5 1 2 3	cbvrexfw	$$⊢\exists x\in A\phi \leftrightarrow \exists y\in A\psi$$