Theorem bj-cbvalvv

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

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

Proof

Step	Hyp	Ref	Expression
1		bj-spvw	$$⊢\exists x\phi \to \left(\psi \leftrightarrow \forall x\psi \right)$$
2	1	biimprd	$$⊢\exists x\phi \to \left(\forall x\psi \to \psi \right)$$
3		ax-5	$$⊢\psi \to \forall y\psi$$
4	2 3	syl6	$$⊢\exists x\phi \to \left(\forall x\psi \to \forall y\psi \right)$$