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Theorem hbnt 2294
Description: Closed theorem version of bound-variable hypothesis builder hbn 2295. (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 2145 . . 3 (∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑)
21nfnd 1858 . 2 (∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥 ¬ 𝜑)
32nf5rd 2196 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 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-10 2141  ax-12 2177
This theorem depends on definitions:  df-bi 207  df-or 849  df-ex 1780  df-nf 1784
This theorem is referenced by:  hbn  2295  hbnd  2296  bj-hbext  36711
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