Theorem cbv3v

Description: Rule used to change bound variables, using implicit substitution. Version of cbv3 with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 5-Aug-1993) (Revised by BJ, 31-May-2019)

	Ref	Expression
Hypotheses	cbv3v.nf1	$$⊢Ⅎy\phi$$
	cbv3v.nf2	$$⊢Ⅎx\psi$$
	cbv3v.1	$$⊢x=y\to \left(\phi \to \psi \right)$$
Assertion	cbv3v	$$⊢\forall x\phi \to \forall y\psi$$

Proof

Step	Hyp	Ref	Expression
1		cbv3v.nf1	$$⊢Ⅎy\phi$$
2		cbv3v.nf2	$$⊢Ⅎx\psi$$
3		cbv3v.1	$$⊢x=y\to \left(\phi \to \psi \right)$$
4	1	nf5ri	$$⊢\phi \to \forall y\phi$$
5	4	hbal	$$⊢\forall x\phi \to \forall y\forall x\phi$$
6	2 3	spimfv	$$⊢\forall x\phi \to \psi$$
7	5 6	alrimih	$$⊢\forall x\phi \to \forall y\psi$$