Theorem cbv3

Description: Rule used to change bound variables, using implicit substitution, that does not use ax-c9 . Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbv3v if possible. (Contributed by NM, 5-Aug-1993) (Proof shortened by Wolf Lammen, 12-May-2018) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cbv3.1	$$⊢Ⅎy\phi$$
	cbv3.2	$$⊢Ⅎx\psi$$
	cbv3.3	$$⊢x=y\to \left(\phi \to \psi \right)$$
Assertion	cbv3	$$⊢\forall x\phi \to \forall y\psi$$

Proof

Step	Hyp	Ref	Expression
1		cbv3.1	$$⊢Ⅎy\phi$$
2		cbv3.2	$$⊢Ⅎx\psi$$
3		cbv3.3	$$⊢x=y\to \left(\phi \to \psi \right)$$
4	1	nf5ri	$$⊢\phi \to \forall y\phi$$
5	4	hbal	$$⊢\forall x\phi \to \forall y\forall x\phi$$
6	2 3	spim	$$⊢\forall x\phi \to \psi$$
7	5 6	alrimih	$$⊢\forall x\phi \to \forall y\psi$$