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Theorem hbn 2131
 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 2129 . 2 (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
2 hbn.1 . 2 (𝜑 → ∀𝑥𝜑)
31, 2mpg 1715 1 𝜑 → ∀𝑥 ¬ 𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1473 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-or 384  df-ex 1696  df-nf 1701 This theorem is referenced by:  hbexOLD  2138  hbnae  2305  ac6s6  33150  hbnae-o  33231  vk15.4j  37755  vk15.4jVD  38172
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