Theorem cbvex2vv

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbvex2vw if possible. (Contributed by NM, 26-Jul-1995) Remove dependency on ax-10 . (Revised by Wolf Lammen, 18-Jul-2021) (New usage is discouraged.)

	Ref	Expression
Hypothesis	cbval2vv.1	$$⊢\left(x=z\wedge y=w\right)\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	cbvex2vv	$$⊢\exists x\exists y\phi \leftrightarrow \exists z\exists w\psi$$

Proof

Step	Hyp	Ref	Expression
1		cbval2vv.1	$$⊢\left(x=z\wedge y=w\right)\to \left(\phi \leftrightarrow \psi \right)$$
2	1	cbvexdva	$$⊢x=z\to \left(\exists y\phi \leftrightarrow \exists w\psi \right)$$
3	2	cbvexv	$$⊢\exists x\exists y\phi \leftrightarrow \exists z\exists w\psi$$