Theorem nftht 1718

Description: Closed form of nfth 1727. (Contributed by Wolf Lammen, 19-Aug-2018.) (Proof shortened by BJ, 16-Sep-2021.)

Assertion

Ref	Expression
nftht	⊢ (∀𝑥𝜑 → Ⅎ𝑥𝜑)

Proof of Theorem nftht

Step	Hyp	Ref	Expression
1		ax-1 6	. 2 ⊢ (∀𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑))
2		df-nf 1710	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
3	1, 2	sylibr 224	1 ⊢ (∀𝑥𝜑 → Ⅎ𝑥𝜑)

Syntax hints:   → wi 4  ∀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-nf 1710

This theorem is referenced by:  nfth  1727  nfim1  2067  wl-nfeqfb  33323