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Theorem nfnt 1619
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 1508 . 2 𝑥𝑥𝜑
2 df-nf 1422 . . 3 (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
3 hbnt 1616 . . 3 (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
42, 3sylbi 120 . 2 (Ⅎ𝑥𝜑 → (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
51, 4nfd 1488 1 (Ⅎ𝑥𝜑 → Ⅎ𝑥 ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wal 1314  wnf 1421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-5 1408  ax-gen 1410  ax-ie2 1455  ax-4 1472  ax-ial 1499
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-fal 1322  df-nf 1422
This theorem is referenced by:  nfnd  1620  nfn  1621
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