Theorem cbvaldw

Description: Deduction used to change bound variables, using implicit substitution. Version of cbvald with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 2-Jan-2002) (Revised by Gino Giotto, 10-Jan-2024)

	Ref	Expression
Hypotheses	cbvaldw.1	$$⊢Ⅎy\phi$$
	cbvaldw.2	$$⊢\phi \to Ⅎy\psi$$
	cbvaldw.3	$$⊢\phi \to \left(x=y\to \left(\psi \leftrightarrow \chi \right)\right)$$
Assertion	cbvaldw	$$⊢\phi \to \left(\forall x\psi \leftrightarrow \forall y\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		cbvaldw.1	$$⊢Ⅎy\phi$$
2		cbvaldw.2	$$⊢\phi \to Ⅎy\psi$$
3		cbvaldw.3	$$⊢\phi \to \left(x=y\to \left(\psi \leftrightarrow \chi \right)\right)$$
4		nfv	$$⊢Ⅎx\phi$$
5		nfvd	$$⊢\phi \to Ⅎx\chi$$
6	4 1 2 5 3	cbv2w	$$⊢\phi \to \left(\forall x\psi \leftrightarrow \forall y\chi \right)$$