Theorem cbvrexsvw

Description: Change bound variable by using a substitution. Version of cbvrexsv with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 2-Mar-2008) (Revised by Gino Giotto, 10-Jan-2024)

	Ref	Expression
Assertion	cbvrexsvw	$$⊢\exists x\in A\phi \leftrightarrow \exists y\in A\left[y/x\right]\phi$$

Proof

Step	Hyp	Ref	Expression
1		nfv	$$⊢Ⅎz\phi$$
2		nfs1v	$$⊢Ⅎx\left[z/x\right]\phi$$
3		sbequ12	$$⊢x=z\to \left(\phi \leftrightarrow \left[z/x\right]\phi \right)$$
4	1 2 3	cbvrexw	$$⊢\exists x\in A\phi \leftrightarrow \exists z\in A\left[z/x\right]\phi$$
5		nfv	$$⊢Ⅎy\left[z/x\right]\phi$$
6		nfv	$$⊢Ⅎz\left[y/x\right]\phi$$
7		sbequ	$$⊢z=y\to \left(\left[z/x\right]\phi \leftrightarrow \left[y/x\right]\phi \right)$$
8	5 6 7	cbvrexw	$$⊢\exists z\in A\left[z/x\right]\phi \leftrightarrow \exists y\in A\left[y/x\right]\phi$$
9	4 8	bitri	$$⊢\exists x\in A\phi \leftrightarrow \exists y\in A\left[y/x\right]\phi$$