Theorem bj-cbvexdv

Description: Version of cbvexd 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-cbvexdv	$$⊢\phi \to \left(\exists x\psi \leftrightarrow \exists y\chi \right)$$

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	2	nfnd	$$⊢\phi \to Ⅎy\neg \psi$$
5		notbi	$$⊢\left(\psi \leftrightarrow \chi \right)\leftrightarrow \left(\neg \psi \leftrightarrow \neg \chi \right)$$
6	3 5	syl6ib	$$⊢\phi \to \left(x=y\to \left(\neg \psi \leftrightarrow \neg \chi \right)\right)$$
7	1 4 6	bj-cbvaldv	$$⊢\phi \to \left(\forall x\neg \psi \leftrightarrow \forall y\neg \chi \right)$$
8	7	notbid	$$⊢\phi \to \left(\neg \forall x\neg \psi \leftrightarrow \neg \forall y\neg \chi \right)$$
9		df-ex	$$⊢\exists x\psi \leftrightarrow \neg \forall x\neg \psi$$
10		df-ex	$$⊢\exists y\chi \leftrightarrow \neg \forall y\neg \chi$$
11	8 9 10	3bitr4g	$$⊢\phi \to \left(\exists x\psi \leftrightarrow \exists y\chi \right)$$