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Theorem or3di 29616
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 1074 . . . 4 ((𝜓𝜒𝜏) ↔ ((𝜓𝜒) ∧ 𝜏))
21orbi2i 542 . . 3 ((𝜑 ∨ (𝜓𝜒𝜏)) ↔ (𝜑 ∨ ((𝜓𝜒) ∧ 𝜏)))
3 ordi 944 . . 3 ((𝜑 ∨ ((𝜓𝜒) ∧ 𝜏)) ↔ ((𝜑 ∨ (𝜓𝜒)) ∧ (𝜑𝜏)))
4 ordi 944 . . . 4 ((𝜑 ∨ (𝜓𝜒)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))
54anbi1i 733 . . 3 (((𝜑 ∨ (𝜓𝜒)) ∧ (𝜑𝜏)) ↔ (((𝜑𝜓) ∧ (𝜑𝜒)) ∧ (𝜑𝜏)))
62, 3, 53bitri 286 . 2 ((𝜑 ∨ (𝜓𝜒𝜏)) ↔ (((𝜑𝜓) ∧ (𝜑𝜒)) ∧ (𝜑𝜏)))
7 df-3an 1074 . 2 (((𝜑𝜓) ∧ (𝜑𝜒) ∧ (𝜑𝜏)) ↔ (((𝜑𝜓) ∧ (𝜑𝜒)) ∧ (𝜑𝜏)))
86, 7bitr4i 267 1 ((𝜑 ∨ (𝜓𝜒𝜏)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒) ∧ (𝜑𝜏)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wo 382  wa 383  w3a 1072
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-an 385  df-3an 1074
This theorem is referenced by:  or3dir  29617
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