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Theorem 3anor 1097
Description: Triple conjunction expressed in terms of triple disjunction. (Contributed by Jeff Hankins, 15-Aug-2009.) (Proof shortened by Wolf Lammen, 8-Apr-2022.)
Assertion
Ref Expression
3anor ((𝜑𝜓𝜒) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒))

Proof of Theorem 3anor
StepHypRef Expression
1 3ianor 1096 . . 3 (¬ (𝜑𝜓𝜒) ↔ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒))
21con1bii 345 . 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 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:  3ianorOLD  1098  ne3anior  3036
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