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Theorem 3ioran 939
Description: Negated triple disjunction as triple conjunction. (Contributed by Scott Fenton, 19-Apr-2011.)
Assertion
Ref Expression
3ioran (¬ (𝜑𝜓𝜒) ↔ (¬ 𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜒))

Proof of Theorem 3ioran
StepHypRef Expression
1 ioran 704 . . 3 (¬ (𝜑𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))
21anbi1i 446 . 2 ((¬ (𝜑𝜓) ∧ ¬ 𝜒) ↔ ((¬ 𝜑 ∧ ¬ 𝜓) ∧ ¬ 𝜒))
3 ioran 704 . . 3 (¬ ((𝜑𝜓) ∨ 𝜒) ↔ (¬ (𝜑𝜓) ∧ ¬ 𝜒))
4 df-3or 925 . . 3 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∨ 𝜒))
53, 4xchnxbir 641 . 2 (¬ (𝜑𝜓𝜒) ↔ (¬ (𝜑𝜓) ∧ ¬ 𝜒))
6 df-3an 926 . 2 ((¬ 𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜒) ↔ ((¬ 𝜑 ∧ ¬ 𝜓) ∧ ¬ 𝜒))
72, 5, 63bitr4i 210 1 (¬ (𝜑𝜓𝜒) ↔ (¬ 𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜒))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 102  wb 103  wo 664  w3o 923  w3a 924
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926
This theorem is referenced by:  ne3anior  2343
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