Theorem nfntht 1710

Description: Closed form of nfnth 1719. (Contributed by BJ, 16-Sep-2021.)

Assertion

Ref	Expression
nfntht	⊢ (¬ ∃𝑥𝜑 → Ⅎ𝑥𝜑)

Proof of Theorem nfntht

Step	Hyp	Ref	Expression
1		olc 398	. 2 ⊢ (¬ ∃𝑥𝜑 → (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
2		nf2 1702	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
3	1, 2	sylibr 223	1 ⊢ (¬ ∃𝑥𝜑 → Ⅎ𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382  ∀wal 1473  ∃wex 1695  Ⅎwnf 1699

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 196  df-or 384  df-nf 1701

This theorem is referenced by: (None)