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	⊢ ( ∃ 𝑥 𝜑 → ( ∃ 𝑦 𝜓 → ∃ 𝑥 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		ax5e	⊢ ( ∃ 𝑦 𝜓 → 𝜓 )
2		ax-5	⊢ ( 𝜓 → ∀ 𝑥 𝜓 )
3	1 2	syl	⊢ ( ∃ 𝑦 𝜓 → ∀ 𝑥 𝜓 )
4		bj-axdd2	⊢ ( ∃ 𝑥 𝜑 → ( ∀ 𝑥 𝜓 → ∃ 𝑥 𝜓 ) )
5	3 4	syl5	⊢ ( ∃ 𝑥 𝜑 → ( ∃ 𝑦 𝜓 → ∃ 𝑥 𝜓 ) )