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Theorem nfnt 1543
Description: If x is not free in φ, then it is not free in ¬ φ. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 28-Dec-2017.) (Revised by BJ, 24-Jul-2019.)
Assertion
Ref Expression
nfnt (Ⅎxφ → Ⅎx ¬ φ)

Proof of Theorem nfnt
StepHypRef Expression
1 nfnf1 1433 . 2 xxφ
2 df-nf 1347 . . 3 (Ⅎxφx(φxφ))
3 hbnt 1540 . . 3 (x(φxφ) → (¬ φx ¬ φ))
42, 3sylbi 114 . 2 (Ⅎxφ → (¬ φx ¬ φ))
51, 4nfd 1413 1 (Ⅎxφ → Ⅎx ¬ φ)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wal 1240  wnf 1346
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1333  ax-gen 1335  ax-ie2 1380  ax-4 1397  ax-ial 1424
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347
This theorem is referenced by:  nfnd  1544  nfn  1545
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