Metamath Proof Explorer


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 ( 𝜑𝜓 ) → ( ∃ 𝑥 𝜑𝜓 ) ) )

Proof

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