Theorem bj-nnf-exlim

Description: Proof of the closed form of exlimi from modalK (compare exlimiv ). See also bj-sylget2 . (Contributed by BJ, 2-Dec-2023)

	Ref	Expression
Assertion	bj-nnf-exlim	⊢ ( Ⅎ' 𝑥 𝜓 → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( ∃ 𝑥 𝜑 → 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		exim	⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( ∃ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) )
2		bj-nnfe	⊢ ( Ⅎ' 𝑥 𝜓 → ( ∃ 𝑥 𝜓 → 𝜓 ) )
3	1 2	syl9r	⊢ ( Ⅎ' 𝑥 𝜓 → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( ∃ 𝑥 𝜑 → 𝜓 ) ) )