Theorem hbnt 2328

Description: Closed theorem version of bound-variable hypothesis builder hbn 2329. (Contributed by NM, 10-May-1993.) (Proof shortened by Wolf Lammen, 14-Oct-2021.)

Assertion

Ref	Expression
hbnt	⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑))

Proof of Theorem hbnt

Step	Hyp	Ref	Expression
1		nf5-1 2179	. . 3 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑)
2	1	nfnd 1887	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥 ¬ 𝜑)
3	2	nf5rd 2231	1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-10 2175  ax-12 2212

This proof depends on definitions:  df-bi 210  df-or 861  df-ex 1809  df-nf 1813

This theorem is used by:  hbn  2329  hbnd  2330