Theorem bj-cbvaldvav

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

	Ref	Expression
Hypothesis	bj-cbvaldvav.1	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \leftrightarrow \chi \right)$$
Assertion	bj-cbvaldvav	$$⊢\phi \to \left(\forall x\psi \leftrightarrow \forall y\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		bj-cbvaldvav.1	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \leftrightarrow \chi \right)$$
2		nfv	$$⊢Ⅎy\phi$$
3		nfvd	$$⊢\phi \to Ⅎy\psi$$
4	1	ex	$$⊢\phi \to \left(x=y\to \left(\psi \leftrightarrow \chi \right)\right)$$
5	2 3 4	bj-cbvaldv	$$⊢\phi \to \left(\forall x\psi \leftrightarrow \forall y\chi \right)$$