Theorem bj-cbvexvv

Description: Existentially quantifying with respect to a non-occurring variable is independent of that variable, over ax-1 -- ax-5 and the existence axiom extru . (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		ax-5	$$⊢\psi \to \forall x\psi$$
3	1 2	syl	$$⊢\exists y\psi \to \forall x\psi$$
4		bj-axdd2	$$⊢\exists x\phi \to \left(\forall x\psi \to \exists x\psi \right)$$
5	3 4	syl5	$$⊢\exists x\phi \to \left(\exists y\psi \to \exists x\psi \right)$$