Theorem nfntht 1719

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

Assertion

Ref	Expression
nfntht	⊢ (¬ ∃𝑥𝜑 → Ⅎ𝑥𝜑)

Proof of Theorem nfntht

Step	Hyp	Ref	Expression
1		olc 399	. 2 ⊢ (¬ ∃𝑥𝜑 → (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
2		nf2 1711	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
3	1, 2	sylibr 224	1 ⊢ (¬ ∃𝑥𝜑 → Ⅎ𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 383  ∀wal 1481  ∃wex 1704  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-nf 1710

This theorem is referenced by: (None)