Theorem nfntht2

Description: Closed form of nfnth . (Contributed by BJ, 16-Sep-2021) (Proof shortened by Wolf Lammen, 4-Sep-2022)

	Ref	Expression
Assertion	nfntht2	⊢ ( ∀ 𝑥 ¬ 𝜑 → Ⅎ 𝑥 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		alnex	⊢ ( ∀ 𝑥 ¬ 𝜑 ↔ ¬ ∃ 𝑥 𝜑 )
2		nfntht	⊢ ( ¬ ∃ 𝑥 𝜑 → Ⅎ 𝑥 𝜑 )
3	1 2	sylbi	⊢ ( ∀ 𝑥 ¬ 𝜑 → Ⅎ 𝑥 𝜑 )