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Theorem 3ioran 988
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 747 . . 3 (¬ (𝜑𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))
21anbi1i 455 . 2 ((¬ (𝜑𝜓) ∧ ¬ 𝜒) ↔ ((¬ 𝜑 ∧ ¬ 𝜓) ∧ ¬ 𝜒))
3 ioran 747 . . 3 (¬ ((𝜑𝜓) ∨ 𝜒) ↔ (¬ (𝜑𝜓) ∧ ¬ 𝜒))
4 df-3or 974 . . 3 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∨ 𝜒))
53, 4xchnxbir 676 . 2 (¬ (𝜑𝜓𝜒) ↔ (¬ (𝜑𝜓) ∧ ¬ 𝜒))
6 df-3an 975 . 2 ((¬ 𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜒) ↔ ((¬ 𝜑 ∧ ¬ 𝜓) ∧ ¬ 𝜒))
72, 5, 63bitr4i 211 1 (¬ (𝜑𝜓𝜒) ↔ (¬ 𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜒))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 103  wb 104  wo 703  w3o 972  w3a 973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975
This theorem is referenced by:  ne3anior  2428  onntri35  7214
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