Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  3or6 Structured version   Visualization version   GIF version

Theorem 3or6 1450
 Description: Analogue of or4 549 for triple conjunction. (Contributed by Scott Fenton, 16-Mar-2011.)
Assertion
Ref Expression
3or6 (((𝜑𝜓) ∨ (𝜒𝜃) ∨ (𝜏𝜂)) ↔ ((𝜑𝜒𝜏) ∨ (𝜓𝜃𝜂)))

Proof of Theorem 3or6
StepHypRef Expression
1 or4 549 . . 3 ((((𝜑𝜒) ∨ 𝜏) ∨ ((𝜓𝜃) ∨ 𝜂)) ↔ (((𝜑𝜒) ∨ (𝜓𝜃)) ∨ (𝜏𝜂)))
2 or4 549 . . . 4 (((𝜑𝜒) ∨ (𝜓𝜃)) ↔ ((𝜑𝜓) ∨ (𝜒𝜃)))
32orbi1i 541 . . 3 ((((𝜑𝜒) ∨ (𝜓𝜃)) ∨ (𝜏𝜂)) ↔ (((𝜑𝜓) ∨ (𝜒𝜃)) ∨ (𝜏𝜂)))
41, 3bitr2i 265 . 2 ((((𝜑𝜓) ∨ (𝜒𝜃)) ∨ (𝜏𝜂)) ↔ (((𝜑𝜒) ∨ 𝜏) ∨ ((𝜓𝜃) ∨ 𝜂)))
5 df-3or 1055 . 2 (((𝜑𝜓) ∨ (𝜒𝜃) ∨ (𝜏𝜂)) ↔ (((𝜑𝜓) ∨ (𝜒𝜃)) ∨ (𝜏𝜂)))
6 df-3or 1055 . . 3 ((𝜑𝜒𝜏) ↔ ((𝜑𝜒) ∨ 𝜏))
7 df-3or 1055 . . 3 ((𝜓𝜃𝜂) ↔ ((𝜓𝜃) ∨ 𝜂))
86, 7orbi12i 542 . 2 (((𝜑𝜒𝜏) ∨ (𝜓𝜃𝜂)) ↔ (((𝜑𝜒) ∨ 𝜏) ∨ ((𝜓𝜃) ∨ 𝜂)))
94, 5, 83bitr4i 292 1 (((𝜑𝜓) ∨ (𝜒𝜃) ∨ (𝜏𝜂)) ↔ ((𝜑𝜒𝜏) ∨ (𝜓𝜃𝜂)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∨ wo 382   ∨ w3o 1053 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-or 384  df-3or 1055 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator