Theorem cbv1h

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 11-May-1993) (Proof shortened by Wolf Lammen, 13-May-2018) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cbv1h.1	$$⊢\phi \to \left(\psi \to \forall y\psi \right)$$
	cbv1h.2	$$⊢\phi \to \left(\chi \to \forall x\chi \right)$$
	cbv1h.3	$$⊢\phi \to \left(x=y\to \left(\psi \to \chi \right)\right)$$
Assertion	cbv1h	$$⊢\forall x\forall y\phi \to \left(\forall x\psi \to \forall y\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		cbv1h.1	$$⊢\phi \to \left(\psi \to \forall y\psi \right)$$
2		cbv1h.2	$$⊢\phi \to \left(\chi \to \forall x\chi \right)$$
3		cbv1h.3	$$⊢\phi \to \left(x=y\to \left(\psi \to \chi \right)\right)$$
4		nfa1	$$⊢Ⅎx\forall x\forall y\phi$$
5		nfa2	$$⊢Ⅎy\forall x\forall y\phi$$
6		2sp	$$⊢\forall x\forall y\phi \to \phi$$
7	6 1	syl	$$⊢\forall x\forall y\phi \to \left(\psi \to \forall y\psi \right)$$
8	5 7	nf5d	$$⊢\forall x\forall y\phi \to Ⅎy\psi$$
9	6 2	syl	$$⊢\forall x\forall y\phi \to \left(\chi \to \forall x\chi \right)$$
10	4 9	nf5d	$$⊢\forall x\forall y\phi \to Ⅎx\chi$$
11	6 3	syl	$$⊢\forall x\forall y\phi \to \left(x=y\to \left(\psi \to \chi \right)\right)$$
12	4 5 8 10 11	cbv1	$$⊢\forall x\forall y\phi \to \left(\forall x\psi \to \forall y\chi \right)$$