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Theorem hbnt 2290
Description: Closed theorem version of bound-variable hypothesis builder hbn 2291. (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
StepHypRef Expression
1 nf5-1 2140 . . 3 (∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑)
21nfnd 1860 . 2 (∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥 ¬ 𝜑)
32nf5rd 2188 1 (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-10 2136  ax-12 2170
This theorem depends on definitions:  df-bi 206  df-or 845  df-ex 1781  df-nf 1785
This theorem is referenced by:  hbn  2291  hbnd  2292  bj-hbext  34988
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