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Theorem hbnt 2291
Description: Closed theorem version of bound-variable hypothesis builder hbn 2293. (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 2172 . . 3 (∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑)
21nfnd 1934 . 2 (∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥 ¬ 𝜑)
32nf5rd 2213 1 (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-10 2168  ax-12 2196
This theorem depends on definitions:  df-bi 197  df-or 384  df-ex 1854  df-nf 1859
This theorem is referenced by:  hbn  2293  hbnd  2294  nfntOLD  2354  bj-hbext  33029
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