Theorem bj-cbvexvv

Description: Existentially quantifying over a non-occurring variable is independent of that variable, over ax-1 -- ax-5 and the existence axiom extru . See bj-cbvew for a strengthening. (Contributed by BJ, 8-Mar-2026) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-cbvexvv	$$⊢\exists x\phi \to \left(\exists y\psi \to \exists x\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		ax5e	$$⊢\exists y\psi \to \psi$$
2		bj-spvew	$$⊢\exists x\phi \to \left(\psi \leftrightarrow \exists x\psi \right)$$
3	2	biimpd	$$⊢\exists x\phi \to \left(\psi \to \exists x\psi \right)$$
4	1 3	syl5	$$⊢\exists x\phi \to \left(\exists y\psi \to \exists x\psi \right)$$