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	⊢ ( ∃ 𝑥 𝜑 → ( ∃ 𝑦 𝜓 → ∃ 𝑥 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		ax5e	⊢ ( ∃ 𝑦 𝜓 → 𝜓 )
2		bj-spvew	⊢ ( ∃ 𝑥 𝜑 → ( 𝜓 ↔ ∃ 𝑥 𝜓 ) )
3	2	biimpd	⊢ ( ∃ 𝑥 𝜑 → ( 𝜓 → ∃ 𝑥 𝜓 ) )
4	1 3	syl5	⊢ ( ∃ 𝑥 𝜑 → ( ∃ 𝑦 𝜓 → ∃ 𝑥 𝜓 ) )