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 ) ) ) |
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 ) ) ) |