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Theorem or3di 30209
Description: Distributive law for disjunction. (Contributed by Thierry Arnoux, 3-Jul-2017.)
Assertion
Ref Expression
or3di ((𝜑 ∨ (𝜓𝜒𝜏)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒) ∧ (𝜑𝜏)))

Proof of Theorem or3di
StepHypRef Expression
1 df-3an 1086 . . . 4 ((𝜓𝜒𝜏) ↔ ((𝜓𝜒) ∧ 𝜏))
21orbi2i 910 . . 3 ((𝜑 ∨ (𝜓𝜒𝜏)) ↔ (𝜑 ∨ ((𝜓𝜒) ∧ 𝜏)))
3 ordi 1003 . . 3 ((𝜑 ∨ ((𝜓𝜒) ∧ 𝜏)) ↔ ((𝜑 ∨ (𝜓𝜒)) ∧ (𝜑𝜏)))
4 ordi 1003 . . . 4 ((𝜑 ∨ (𝜓𝜒)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))
54anbi1i 626 . . 3 (((𝜑 ∨ (𝜓𝜒)) ∧ (𝜑𝜏)) ↔ (((𝜑𝜓) ∧ (𝜑𝜒)) ∧ (𝜑𝜏)))
62, 3, 53bitri 300 . 2 ((𝜑 ∨ (𝜓𝜒𝜏)) ↔ (((𝜑𝜓) ∧ (𝜑𝜒)) ∧ (𝜑𝜏)))
7 df-3an 1086 . 2 (((𝜑𝜓) ∧ (𝜑𝜒) ∧ (𝜑𝜏)) ↔ (((𝜑𝜓) ∧ (𝜑𝜒)) ∧ (𝜑𝜏)))
86, 7bitr4i 281 1 ((𝜑 ∨ (𝜓𝜒𝜏)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒) ∧ (𝜑𝜏)))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wo 844  w3a 1084
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 210  df-an 400  df-or 845  df-3an 1086
This theorem is referenced by:  or3dir  30210
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