ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  3or6 GIF version

Theorem 3or6 1229
Description: Analog of or4 698 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 698 . . 3 ((((𝜑𝜒) ∨ 𝜏) ∨ ((𝜓𝜃) ∨ 𝜂)) ↔ (((𝜑𝜒) ∨ (𝜓𝜃)) ∨ (𝜏𝜂)))
2 or4 698 . . . 4 (((𝜑𝜒) ∨ (𝜓𝜃)) ↔ ((𝜑𝜓) ∨ (𝜒𝜃)))
32orbi1i 690 . . 3 ((((𝜑𝜒) ∨ (𝜓𝜃)) ∨ (𝜏𝜂)) ↔ (((𝜑𝜓) ∨ (𝜒𝜃)) ∨ (𝜏𝜂)))
41, 3bitr2i 178 . 2 ((((𝜑𝜓) ∨ (𝜒𝜃)) ∨ (𝜏𝜂)) ↔ (((𝜑𝜒) ∨ 𝜏) ∨ ((𝜓𝜃) ∨ 𝜂)))
5 df-3or 897 . 2 (((𝜑𝜓) ∨ (𝜒𝜃) ∨ (𝜏𝜂)) ↔ (((𝜑𝜓) ∨ (𝜒𝜃)) ∨ (𝜏𝜂)))
6 df-3or 897 . . 3 ((𝜑𝜒𝜏) ↔ ((𝜑𝜒) ∨ 𝜏))
7 df-3or 897 . . 3 ((𝜓𝜃𝜂) ↔ ((𝜓𝜃) ∨ 𝜂))
86, 7orbi12i 691 . 2 (((𝜑𝜒𝜏) ∨ (𝜓𝜃𝜂)) ↔ (((𝜑𝜒) ∨ 𝜏) ∨ ((𝜓𝜃) ∨ 𝜂)))
94, 5, 83bitr4i 205 1 (((𝜑𝜓) ∨ (𝜒𝜃) ∨ (𝜏𝜂)) ↔ ((𝜑𝜒𝜏) ∨ (𝜓𝜃𝜂)))
Colors of variables: wff set class
Syntax hints:  wb 102  wo 639  w3o 895
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640
This theorem depends on definitions:  df-bi 114  df-3or 897
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator