Theorem cbval2vv

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 cbval2vw if possible. (Contributed by NM, 4-Feb-2005) 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	cbval2vv	$$⊢\forall x\forall y\phi \leftrightarrow \forall z\forall w\psi$$

Proof

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