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Theorem 3ianor 1054
 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.)
Assertion
Ref Expression
3ianor (¬ (𝜑𝜓𝜒) ↔ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒))

Proof of Theorem 3ianor
StepHypRef Expression
1 3anor 1053 . . 3 ((𝜑𝜓𝜒) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒))
21con2bii 347 . 2 ((¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒) ↔ ¬ (𝜑𝜓𝜒))
32bicomi 214 1 (¬ (𝜑𝜓𝜒) ↔ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 196   ∨ w3o 1036   ∧ w3a 1037 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-or 385  df-an 386  df-3or 1038  df-3an 1039 This theorem is referenced by:  tppreqb  4334  funtpgOLD  5941  fr3nr  6976  bropopvvv  7252  prinfzo0  12502  elfznelfzo  12569  ssnn0fi  12779  hashtpg  13262  lcmfunsnlem2lem2  15346  prm23ge5  15514  lpni  27316  xrdifh  29527  dvasin  33476  limcicciooub  39675  2zrngnring  41723
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