Theorem nfntht2 1822

Description: Closed form of nfnth 1830. (Contributed by BJ, 16-Sep-2021.) (Proof shortened by Wolf Lammen, 4-Sep-2022.)

Assertion

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

Proof of Theorem nfntht2

Step	Hyp	Ref	Expression
1		alnex 1809	. 2 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
2		nfntht 1821	. 2 ⊢ (¬ ∃𝑥𝜑 → Ⅎ𝑥𝜑)
3	1, 2	sylbi 220	1 ⊢ (∀𝑥 ¬ 𝜑 → Ⅎ𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1566  ∃wex 1807  Ⅎwnf 1811

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 210  df-ex 1808  df-nf 1812

This theorem is referenced by:  nfnth  1830