Theorem cbval2vw

Description: Rule used to change bound variables, using implicit substitution. Version of cbval2vv with more disjoint variable conditions, which requires fewer axioms . (Contributed by NM, 4-Feb-2005) 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	cbval2vw	$$⊢\forall x\forall y\phi \leftrightarrow \forall z\forall w\psi$$

Proof

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