Theorem cbvex2vw

Description: Rule used to change bound variables, using implicit substitution. Version of cbvex2vv with more disjoint variable conditions, which requires fewer axioms . (Contributed by NM, 26-Jul-1995) Avoid ax-13 . (Revised by Gino Giotto, 10-Jan-2024)

	Ref	Expression
Hypothesis	cbval2vw.1	$$⊢\left(x=z\wedge y=w\right)\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	cbvex2vw	$$⊢\exists x\exists y\phi \leftrightarrow \exists z\exists w\psi$$

Proof

Step	Hyp	Ref	Expression
1		cbval2vw.1	$$⊢\left(x=z\wedge y=w\right)\to \left(\phi \leftrightarrow \psi \right)$$
2	1	cbvexdvaw	$$⊢x=z\to \left(\exists y\phi \leftrightarrow \exists w\psi \right)$$
3	2	cbvexvw	$$⊢\exists x\exists y\phi \leftrightarrow \exists z\exists w\psi$$