Theorem bj-cbvaldv

Description: Version of cbvald 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-cbvaldv.1	$$⊢Ⅎy\phi$$
	bj-cbvaldv.2	$$⊢\phi \to Ⅎy\psi$$
	bj-cbvaldv.3	$$⊢\phi \to \left(x=y\to \left(\psi \leftrightarrow \chi \right)\right)$$
Assertion	bj-cbvaldv	$$⊢\phi \to \left(\forall x\psi \leftrightarrow \forall y\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		bj-cbvaldv.1	$$⊢Ⅎy\phi$$
2		bj-cbvaldv.2	$$⊢\phi \to Ⅎy\psi$$
3		bj-cbvaldv.3	$$⊢\phi \to \left(x=y\to \left(\psi \leftrightarrow \chi \right)\right)$$
4		nfv	$$⊢Ⅎx\phi$$
5		nfv	$$⊢Ⅎx\chi$$
6	5	a1i	$$⊢\phi \to Ⅎx\chi$$
7	4 1 2 6 3	bj-cbv2v	$$⊢\phi \to \left(\forall x\psi \leftrightarrow \forall y\chi \right)$$