Theorem cbvexv1

Description: Rule used to change bound variables, using implicit substitution. Version of cbvex with a disjoint variable condition, which does not require ax-13 . See cbvexvw for a version with two disjoint variable conditions, requiring fewer axioms, and cbvexv for another variant. (Contributed by NM, 21-Jun-1993) (Revised by BJ, 31-May-2019)

	Ref	Expression
Hypotheses	cbvalv1.nf1	$$⊢Ⅎy\phi$$
	cbvalv1.nf2	$$⊢Ⅎx\psi$$
	cbvalv1.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	cbvexv1	$$⊢\exists x\phi \leftrightarrow \exists y\psi$$

Proof

Step	Hyp	Ref	Expression
1		cbvalv1.nf1	$$⊢Ⅎy\phi$$
2		cbvalv1.nf2	$$⊢Ⅎx\psi$$
3		cbvalv1.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
4	1	nfn	$$⊢Ⅎy\neg \phi$$
5	2	nfn	$$⊢Ⅎx\neg \psi$$
6	3	notbid	$$⊢x=y\to \left(\neg \phi \leftrightarrow \neg \psi \right)$$
7	4 5 6	cbvalv1	$$⊢\forall x\neg \phi \leftrightarrow \forall y\neg \psi$$
8		alnex	$$⊢\forall x\neg \phi \leftrightarrow \neg \exists x\phi$$
9		alnex	$$⊢\forall y\neg \psi \leftrightarrow \neg \exists y\psi$$
10	7 8 9	3bitr3i	$$⊢\neg \exists x\phi \leftrightarrow \neg \exists y\psi$$
11	10	con4bii	$$⊢\exists x\phi \leftrightarrow \exists y\psi$$