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Theorem nfnt 1767
 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.) df-nf 1701 changed. (Revised by Wolf Lammen, 4-Oct-2021.)
Assertion
Ref Expression
nfnt (Ⅎ𝑥𝜑 → Ⅎ𝑥 ¬ 𝜑)

Proof of Theorem nfnt
StepHypRef Expression
1 notnot 135 . . . . 5 (𝜑 → ¬ ¬ 𝜑)
21alimi 1730 . . . 4 (∀𝑥𝜑 → ∀𝑥 ¬ ¬ 𝜑)
32orim1i 538 . . 3 ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) → (∀𝑥 ¬ ¬ 𝜑 ∨ ∀𝑥 ¬ 𝜑))
4 pm1.4 400 . . 3 ((∀𝑥 ¬ ¬ 𝜑 ∨ ∀𝑥 ¬ 𝜑) → (∀𝑥 ¬ 𝜑 ∨ ∀𝑥 ¬ ¬ 𝜑))
53, 4syl 17 . 2 ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) → (∀𝑥 ¬ 𝜑 ∨ ∀𝑥 ¬ ¬ 𝜑))
6 nf3 1703 . 2 (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))
7 nf3 1703 . 2 (Ⅎ𝑥 ¬ 𝜑 ↔ (∀𝑥 ¬ 𝜑 ∨ ∀𝑥 ¬ ¬ 𝜑))
85, 6, 73imtr4i 280 1 (Ⅎ𝑥𝜑 → Ⅎ𝑥 ¬ 𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382  ∀wal 1473  Ⅎwnf 1699 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728 This theorem depends on definitions:  df-bi 196  df-or 384  df-ex 1696  df-nf 1701 This theorem is referenced by:  nfn  1768  nfnd  1769  nfan1  2056  19.23t  2066  wl-nfnbi  32493  wl-nfeqfb  32502  19.9alt  33270
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