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Theorem 3ioran 1094
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 912 . . 3 (¬ (𝜑𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))
21anbi1i 602 . 2 ((¬ (𝜑𝜓) ∧ ¬ 𝜒) ↔ ((¬ 𝜑 ∧ ¬ 𝜓) ∧ ¬ 𝜒))
3 ioran 912 . . 3 (¬ ((𝜑𝜓) ∨ 𝜒) ↔ (¬ (𝜑𝜓) ∧ ¬ 𝜒))
4 df-3or 1071 . . 3 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∨ 𝜒))
53, 4xchnxbir 322 . 2 (¬ (𝜑𝜓𝜒) ↔ (¬ (𝜑𝜓) ∧ ¬ 𝜒))
6 df-3an 1072 . 2 ((¬ 𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜒) ↔ ((¬ 𝜑 ∧ ¬ 𝜓) ∧ ¬ 𝜒))
72, 5, 63bitr4i 292 1 (¬ (𝜑𝜓𝜒) ↔ (¬ 𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 382  wo 826  w3o 1069  w3a 1070
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 827  df-3or 1071  df-3an 1072
This theorem is referenced by:  3oran  1098  cadnot  1701  lcmftp  15556  prm23ge5  15726  cnfldfunALT  19973  fbunfip  21892  frgrregord013  27588  wl-nfeqfb  33651
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