Theorem bj-cbvexw

Description: Change bound variable. This is to cbvexvw what cbvalw is to cbvalvw . (Contributed by BJ, 17-Mar-2020)

	Ref	Expression
Hypotheses	bj-cbvexw.1	$$⊢\exists x\exists y\psi \to \exists y\psi$$
	bj-cbvexw.2	$$⊢\phi \to \forall y\phi$$
	bj-cbvexw.3	$$⊢\exists y\exists x\phi \to \exists x\phi$$
	bj-cbvexw.4	$$⊢\psi \to \forall x\psi$$
	bj-cbvexw.5	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	bj-cbvexw	$$⊢\exists x\phi \leftrightarrow \exists y\psi$$

Proof

Step	Hyp	Ref	Expression
1		bj-cbvexw.1	$$⊢\exists x\exists y\psi \to \exists y\psi$$
2		bj-cbvexw.2	$$⊢\phi \to \forall y\phi$$
3		bj-cbvexw.3	$$⊢\exists y\exists x\phi \to \exists x\phi$$
4		bj-cbvexw.4	$$⊢\psi \to \forall x\psi$$
5		bj-cbvexw.5	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
6	5	equcoms	$$⊢y=x\to \left(\phi \leftrightarrow \psi \right)$$
7	6	biimpd	$$⊢y=x\to \left(\phi \to \psi \right)$$
8	1 2 7	bj-cbvexiw	$$⊢\exists x\phi \to \exists y\psi$$
9	5	biimprd	$$⊢x=y\to \left(\psi \to \phi \right)$$
10	3 4 9	bj-cbvexiw	$$⊢\exists y\psi \to \exists x\phi$$
11	8 10	impbii	$$⊢\exists x\phi \leftrightarrow \exists y\psi$$