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Theorem 3ianor 1096
Description: Negated triple conjunction expressed in terms of triple disjunction. (Contributed by Jeff Hankins, 15-Aug-2009.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Revised by Wolf Lammen, 8-Apr-2022.)
Assertion
Ref Expression
3ianor (¬ (𝜑𝜓𝜒) ↔ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒))

Proof of Theorem 3ianor
StepHypRef Expression
1 ianor 966 . . 3 (¬ (𝜑𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓))
21orbi1i 899 . 2 ((¬ (𝜑𝜓) ∨ ¬ 𝜒) ↔ ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ 𝜒))
3 ianor 966 . . 3 (¬ ((𝜑𝜓) ∧ 𝜒) ↔ (¬ (𝜑𝜓) ∨ ¬ 𝜒))
4 df-3an 1073 . . 3 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))
53, 4xchnxbir 322 . 2 (¬ (𝜑𝜓𝜒) ↔ (¬ (𝜑𝜓) ∨ ¬ 𝜒))
6 df-3or 1072 . 2 ((¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒) ↔ ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ 𝜒))
72, 5, 63bitr4i 292 1 (¬ (𝜑𝜓𝜒) ↔ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 382  wo 836  w3o 1070  w3a 1071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073
This theorem is referenced by:  3anor  1097  tppreqb  4471  fr3nr  7126  bropopvvv  7406  prinfzo0  12715  elfznelfzo  12781  ssnn0fi  12992  hashtpg  13469  lcmfunsnlem2lem2  15560  prm23ge5  15727  lpni  27674  xrdifh  29882  dvasin  33828  limcicciooub  40387  2zrngnring  42480
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