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Theorem hbn 2311
Description: If 𝑥 is not free in 𝜑, it is not free in ¬ 𝜑. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Wolf Lammen, 17-Dec-2017.)
Hypothesis
Ref Expression
hbn.1 (𝜑 → ∀𝑥𝜑)
Assertion
Ref Expression
hbn 𝜑 → ∀𝑥 ¬ 𝜑)

Proof of Theorem hbn
StepHypRef Expression
1 hbnt 2309 . 2 (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
2 hbn.1 . 2 (𝜑 → ∀𝑥𝜑)
31, 2mpg 1872 1 𝜑 → ∀𝑥 ¬ 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-10 2174  ax-12 2203
This theorem depends on definitions:  df-bi 197  df-or 837  df-ex 1853  df-nf 1858
This theorem is referenced by:  hbexOLD  2316  hbnae  2469  ac6s6  34305  hbnae-o  34729  vk15.4j  39252  vk15.4jVD  39665
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