Theorem nfntht2 1720

Description: Closed form of nfnth 1728. (Contributed by BJ, 16-Sep-2021.)

Assertion

Ref	Expression
nfntht2	⊢ (∀𝑥 ¬ 𝜑 → Ⅎ𝑥𝜑)

Proof of Theorem nfntht2

Step	Hyp	Ref	Expression
1		olc 399	. 2 ⊢ (∀𝑥 ¬ 𝜑 → (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))
2		nf3 1712	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))
3	1, 2	sylibr 224	1 ⊢ (∀𝑥 ¬ 𝜑 → Ⅎ𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 383  ∀wal 1481  Ⅎwnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1705  df-nf 1710

This theorem is referenced by:  nfnth  1728