Theorem bj-cbv1hv

Description: Version of cbv1h with a disjoint variable condition, which does not require ax-13 . (Contributed by BJ, 16-Jun-2019) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	bj-cbv1hv.1	$$⊢\phi \to \left(\psi \to \forall y\psi \right)$$
	bj-cbv1hv.2	$$⊢\phi \to \left(\chi \to \forall x\chi \right)$$
	bj-cbv1hv.3	$$⊢\phi \to \left(x=y\to \left(\psi \to \chi \right)\right)$$
Assertion	bj-cbv1hv	$$⊢\forall x\forall y\phi \to \left(\forall x\psi \to \forall y\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		bj-cbv1hv.1	$$⊢\phi \to \left(\psi \to \forall y\psi \right)$$
2		bj-cbv1hv.2	$$⊢\phi \to \left(\chi \to \forall x\chi \right)$$
3		bj-cbv1hv.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	cbv1v	$$⊢\forall x\forall y\phi \to \left(\forall x\psi \to \forall y\chi \right)$$