Theorem cbvalv

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . See cbvalvw for a version requiring fewer axioms, to be preferred when sufficient. (Contributed by NM, 5-Aug-1993) Remove dependency on ax-10 , shorten. (Revised by Wolf Lammen, 11-Sep-2023) (New usage is discouraged.)

	Ref	Expression
Hypothesis	cbvalv.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	cbvalv	$$⊢\forall x\phi \leftrightarrow \forall y\psi$$

Proof

Step	Hyp	Ref	Expression
1		cbvalv.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
2		nfv	$$⊢Ⅎy\phi$$
3		nfv	$$⊢Ⅎx\psi$$
4	2 3 1	cbval	$$⊢\forall x\phi \leftrightarrow \forall y\psi$$