Theorem cbv3h

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 cbv3hv if possible. (Contributed by NM, 8-Jun-1993) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof shortened by Wolf Lammen, 12-May-2018) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cbv3h.1	$$⊢\phi \to \forall y\phi$$
	cbv3h.2	$$⊢\psi \to \forall x\psi$$
	cbv3h.3	$$⊢x=y\to \left(\phi \to \psi \right)$$
Assertion	cbv3h	$$⊢\forall x\phi \to \forall y\psi$$

Proof

Step	Hyp	Ref	Expression
1		cbv3h.1	$$⊢\phi \to \forall y\phi$$
2		cbv3h.2	$$⊢\psi \to \forall x\psi$$
3		cbv3h.3	$$⊢x=y\to \left(\phi \to \psi \right)$$
4	1	nf5i	$$⊢Ⅎy\phi$$
5	2	nf5i	$$⊢Ⅎx\psi$$
6	4 5 3	cbv3	$$⊢\forall x\phi \to \forall y\psi$$