Theorem cbv3hv

Description: Rule used to change bound variables, using implicit substitution. Version of cbv3h with a disjoint variable condition on x , y , which does not require ax-13 . Was used in a proof of axc11n (but of independent interest). (Contributed by NM, 25-Jul-2015) (Proof shortened by Wolf Lammen, 29-Nov-2020) (Proof shortened by BJ, 30-Nov-2020)

	Ref	Expression
Hypotheses	cbv3hv.nf1	$$⊢\phi \to \forall y\phi$$
	cbv3hv.nf2	$$⊢\psi \to \forall x\psi$$
	cbv3hv.1	$$⊢x=y\to \left(\phi \to \psi \right)$$
Assertion	cbv3hv	$$⊢\forall x\phi \to \forall y\psi$$

Proof

Step	Hyp	Ref	Expression
1		cbv3hv.nf1	$$⊢\phi \to \forall y\phi$$
2		cbv3hv.nf2	$$⊢\psi \to \forall x\psi$$
3		cbv3hv.1	$$⊢x=y\to \left(\phi \to \psi \right)$$
4	1	nf5i	$$⊢Ⅎy\phi$$
5	2	nf5i	$$⊢Ⅎx\psi$$
6	4 5 3	cbv3v	$$⊢\forall x\phi \to \forall y\psi$$