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Theorem 3or6 1259
Description: Analog of or4 723 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 723 . . 3 ((((𝜑𝜒) ∨ 𝜏) ∨ ((𝜓𝜃) ∨ 𝜂)) ↔ (((𝜑𝜒) ∨ (𝜓𝜃)) ∨ (𝜏𝜂)))
2 or4 723 . . . 4 (((𝜑𝜒) ∨ (𝜓𝜃)) ↔ ((𝜑𝜓) ∨ (𝜒𝜃)))
32orbi1i 715 . . 3 ((((𝜑𝜒) ∨ (𝜓𝜃)) ∨ (𝜏𝜂)) ↔ (((𝜑𝜓) ∨ (𝜒𝜃)) ∨ (𝜏𝜂)))
41, 3bitr2i 183 . 2 ((((𝜑𝜓) ∨ (𝜒𝜃)) ∨ (𝜏𝜂)) ↔ (((𝜑𝜒) ∨ 𝜏) ∨ ((𝜓𝜃) ∨ 𝜂)))
5 df-3or 925 . 2 (((𝜑𝜓) ∨ (𝜒𝜃) ∨ (𝜏𝜂)) ↔ (((𝜑𝜓) ∨ (𝜒𝜃)) ∨ (𝜏𝜂)))
6 df-3or 925 . . 3 ((𝜑𝜒𝜏) ↔ ((𝜑𝜒) ∨ 𝜏))
7 df-3or 925 . . 3 ((𝜓𝜃𝜂) ↔ ((𝜓𝜃) ∨ 𝜂))
86, 7orbi12i 716 . 2 (((𝜑𝜒𝜏) ∨ (𝜓𝜃𝜂)) ↔ (((𝜑𝜒) ∨ 𝜏) ∨ ((𝜓𝜃) ∨ 𝜂)))
94, 5, 83bitr4i 210 1 (((𝜑𝜓) ∨ (𝜒𝜃) ∨ (𝜏𝜂)) ↔ ((𝜑𝜒𝜏) ∨ (𝜓𝜃𝜂)))
Colors of variables: wff set class
Syntax hints:  wb 103  wo 664  w3o 923
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665
This theorem depends on definitions:  df-bi 115  df-3or 925
This theorem is referenced by: (None)
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