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Theorem hbnt 2129
 Description: Closed theorem version of bound-variable hypothesis builder hbn 2131. (Contributed by NM, 10-May-1993.) (Proof shortened by Wolf Lammen, 3-Mar-2018.) (Proof shortened by Wolf Lammen, 14-Oct-2021.)
Assertion
Ref Expression
hbnt (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑))

Proof of Theorem hbnt
StepHypRef Expression
1 nf5-1 2010 . . 3 (∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑)
21nfnd 1769 . 2 (∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥 ¬ 𝜑)
32nf5rd 2054 1 (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1473 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-or 384  df-ex 1696  df-nf 1701 This theorem is referenced by:  hbn  2131  hbnd  2132  nfntOLD  2197  bj-hbext  31888
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