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Theorem nfntOLDOLD 1823
 Description: Obsolete proof of nfnt 1822 as of 3-Nov-2021. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 28-Dec-2017.) (Revised by BJ, 24-Jul-2019.) df-nf 1750 changed. (Revised by Wolf Lammen, 4-Oct-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
nfntOLDOLD (Ⅎ𝑥𝜑 → Ⅎ𝑥 ¬ 𝜑)

Proof of Theorem nfntOLDOLD
StepHypRef Expression
1 notnot 136 . . . . 5 (𝜑 → ¬ ¬ 𝜑)
21alimi 1779 . . . 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 1752 . 2 (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))
7 nf3 1752 . 2 (Ⅎ𝑥 ¬ 𝜑 ↔ (∀𝑥 ¬ 𝜑 ∨ ∀𝑥 ¬ ¬ 𝜑))
85, 6, 73imtr4i 281 1 (Ⅎ𝑥𝜑 → Ⅎ𝑥 ¬ 𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382  ∀wal 1521  Ⅎwnf 1748 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777 This theorem depends on definitions:  df-bi 197  df-or 384  df-ex 1745  df-nf 1750 This theorem is referenced by: (None)
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