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Theorem 3ianor 1106
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 983 . . 3 (¬ (𝜑𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓))
21orbi1i 913 . 2 ((¬ (𝜑𝜓) ∨ ¬ 𝜒) ↔ ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ 𝜒))
3 ianor 983 . . 3 (¬ ((𝜑𝜓) ∧ 𝜒) ↔ (¬ (𝜑𝜓) ∨ ¬ 𝜒))
4 df-3an 1088 . . 3 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))
53, 4xchnxbir 333 . 2 (¬ (𝜑𝜓𝜒) ↔ (¬ (𝜑𝜓) ∨ ¬ 𝜒))
6 df-3or 1087 . 2 ((¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒) ↔ ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ 𝜒))
72, 5, 63bitr4i 303 1 (¬ (𝜑𝜓𝜒) ↔ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wa 395  wo 847  w3o 1085  w3a 1086
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 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088
This theorem is referenced by:  3anor  1107  tppreqb  4810  otthne  5497  fr3nr  7791  bropopvvv  8114  prinfzo0  13735  elfznelfzo  13808  ssnn0fi  14023  hashtpg  14521  hash3tpde  14529  swrdnd0  14692  pfxnd0  14723  lcmfunsnlem2lem2  16673  prm23ge5  16849  2irrexpq  26788  lpni  30509  xrdifh  32789  dvasin  37691  dflim5  43319  limcicciooub  45593  2zrngnring  48102
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