Theorem nftht0 1709

Description: Closed form of nfth 1718. (Contributed by Wolf Lammen, 19-Aug-2018.) (Proof shortened by BJ, 16-Sep-2021.)

Assertion

Ref	Expression
nftht0	⊢ (∀𝑥𝜑 → Ⅎ𝑥𝜑)

Proof of Theorem nftht0

Step	Hyp	Ref	Expression
1		ax-1 6	. 2 ⊢ (∀𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑))
2		df-nf 1701	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
3	1, 2	sylibr 223	1 ⊢ (∀𝑥𝜑 → Ⅎ𝑥𝜑)

Syntax hints:   → wi 4  ∀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-nf 1701

This theorem is referenced by:  nfth  1718  nfim1  2055  wl-nfeqfb  32502