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Theorem or3dir 30141
Description: Distributive law for disjunction. (Contributed by Thierry Arnoux, 3-Jul-2017.)
Assertion
Ref Expression
or3dir (((𝜑𝜓𝜒) ∨ 𝜏) ↔ ((𝜑𝜏) ∧ (𝜓𝜏) ∧ (𝜒𝜏)))

Proof of Theorem or3dir
StepHypRef Expression
1 or3di 30140 . 2 ((𝜏 ∨ (𝜑𝜓𝜒)) ↔ ((𝜏𝜑) ∧ (𝜏𝜓) ∧ (𝜏𝜒)))
2 orcom 866 . 2 ((𝜏 ∨ (𝜑𝜓𝜒)) ↔ ((𝜑𝜓𝜒) ∨ 𝜏))
3 orcom 866 . . 3 ((𝜏𝜑) ↔ (𝜑𝜏))
4 orcom 866 . . 3 ((𝜏𝜓) ↔ (𝜓𝜏))
5 orcom 866 . . 3 ((𝜏𝜒) ↔ (𝜒𝜏))
63, 4, 53anbi123i 1149 . 2 (((𝜏𝜑) ∧ (𝜏𝜓) ∧ (𝜏𝜒)) ↔ ((𝜑𝜏) ∧ (𝜓𝜏) ∧ (𝜒𝜏)))
71, 2, 63bitr3i 302 1 (((𝜑𝜓𝜒) ∨ 𝜏) ↔ ((𝜑𝜏) ∧ (𝜓𝜏) ∧ (𝜒𝜏)))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wo 843  w3a 1081
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 208  df-an 397  df-or 844  df-3an 1083
This theorem is referenced by: (None)
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