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Theorem hbn 1654
Description: If 𝑥 is not free in 𝜑, it is not free in ¬ 𝜑. (Contributed by NM, 5-Aug-1993.)
Hypothesis
Ref Expression
hbn.1 (𝜑 → ∀𝑥𝜑)
Assertion
Ref Expression
hbn 𝜑 → ∀𝑥 ¬ 𝜑)

Proof of Theorem hbn
StepHypRef Expression
1 hbnt 1653 . 2 (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
2 hbn.1 . 2 (𝜑 → ∀𝑥𝜑)
31, 2mpg 1451 1 𝜑 → ∀𝑥 ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wal 1351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-5 1447  ax-gen 1449  ax-ie2 1494  ax-4 1510  ax-ial 1534
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359
This theorem is referenced by:  hbnae  1721  sbn  1952  euor  2052  euor2  2084
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