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Theorem 3ioran 1049
 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 510 . . 3 (¬ (𝜑𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))
21anbi1i 727 . 2 ((¬ (𝜑𝜓) ∧ ¬ 𝜒) ↔ ((¬ 𝜑 ∧ ¬ 𝜓) ∧ ¬ 𝜒))
3 ioran 510 . . 3 (¬ ((𝜑𝜓) ∨ 𝜒) ↔ (¬ (𝜑𝜓) ∧ ¬ 𝜒))
4 df-3or 1032 . . 3 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∨ 𝜒))
53, 4xchnxbir 322 . 2 (¬ (𝜑𝜓𝜒) ↔ (¬ (𝜑𝜓) ∧ ¬ 𝜒))
6 df-3an 1033 . 2 ((¬ 𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜒) ↔ ((¬ 𝜑 ∧ ¬ 𝜓) ∧ ¬ 𝜒))
72, 5, 63bitr4i 291 1 (¬ (𝜑𝜓𝜒) ↔ (¬ 𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜒))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∨ w3o 1030   ∧ w3a 1031 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 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033 This theorem is referenced by:  3oran  1050  cadnot  1545  lcmftp  15187  prm23ge5  15358  fbunfip  21483  wwlknndef  26265  wwlknfi  26266  clwwlknndef  26301  frgraregord013  26645  wl-nfeqfb  32502  av-frgraregord013  41549
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