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	⊢ ( ∃ 𝑥 𝜑 → ( ∀ 𝑥 𝜓 → ∀ 𝑦 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		bj-spvw	⊢ ( ∃ 𝑥 𝜑 → ( 𝜓 ↔ ∀ 𝑥 𝜓 ) )
2	1	biimprd	⊢ ( ∃ 𝑥 𝜑 → ( ∀ 𝑥 𝜓 → 𝜓 ) )
3		ax-5	⊢ ( 𝜓 → ∀ 𝑦 𝜓 )
4	2 3	syl6	⊢ ( ∃ 𝑥 𝜑 → ( ∀ 𝑥 𝜓 → ∀ 𝑦 𝜓 ) )