Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  3anor Structured version   Visualization version   GIF version

Theorem 3anor 1047
 Description: Triple conjunction expressed in terms of triple disjunction. (Contributed by Jeff Hankins, 15-Aug-2009.)
Assertion
Ref Expression
3anor ((𝜑𝜓𝜒) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒))

Proof of Theorem 3anor
StepHypRef Expression
1 df-3an 1033 . 2 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))
2 anor 509 . . . 4 (((𝜑𝜓) ∧ 𝜒) ↔ ¬ (¬ (𝜑𝜓) ∨ ¬ 𝜒))
3 ianor 508 . . . . 5 (¬ (𝜑𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓))
43orbi1i 541 . . . 4 ((¬ (𝜑𝜓) ∨ ¬ 𝜒) ↔ ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ 𝜒))
52, 4xchbinx 323 . . 3 (((𝜑𝜓) ∧ 𝜒) ↔ ¬ ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ 𝜒))
6 df-3or 1032 . . 3 ((¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒) ↔ ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ 𝜒))
75, 6xchbinxr 324 . 2 (((𝜑𝜓) ∧ 𝜒) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒))
81, 7bitri 263 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:  3ianor  1048  ne3anior  2875
 Copyright terms: Public domain W3C validator