MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hbnt Structured version   Visualization version   GIF version

Theorem hbnt 2293
Description: Closed theorem version of bound-variable hypothesis builder hbn 2294. (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 2143 . . 3 (∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑)
21nfnd 1856 . 2 (∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥 ¬ 𝜑)
32nf5rd 2194 1 (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-10 2139  ax-12 2175
This theorem depends on definitions:  df-bi 207  df-or 848  df-ex 1777  df-nf 1781
This theorem is referenced by:  hbn  2294  hbnd  2295  bj-hbext  36693
  Copyright terms: Public domain W3C validator