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Theorem ordi 1028
Description: Distributive law for disjunction. Theorem *4.41 of [WhiteheadRussell] p. 119. (Contributed by NM, 5-Jan-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 28-Nov-2013.)
Assertion
Ref Expression
ordi ((𝜑 ∨ (𝜓𝜒)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))

Proof of Theorem ordi
StepHypRef Expression
1 jcab 513 . 2 ((¬ 𝜑 → (𝜓𝜒)) ↔ ((¬ 𝜑𝜓) ∧ (¬ 𝜑𝜒)))
2 df-or 874 . 2 ((𝜑 ∨ (𝜓𝜒)) ↔ (¬ 𝜑 → (𝜓𝜒)))
3 df-or 874 . . 3 ((𝜑𝜓) ↔ (¬ 𝜑𝜓))
4 df-or 874 . . 3 ((𝜑𝜒) ↔ (¬ 𝜑𝜒))
53, 4anbi12i 620 . 2 (((𝜑𝜓) ∧ (𝜑𝜒)) ↔ ((¬ 𝜑𝜓) ∧ (¬ 𝜑𝜒)))
61, 2, 53bitr4i 294 1 ((𝜑 ∨ (𝜓𝜒)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 873
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 198  df-an 385  df-or 874
This theorem is referenced by:  ordir  1029  orddi  1032  pm5.63  1043  pm4.43  1046  cadan  1718  undi  4039  undif3  4053  undif4  4195  elnn1uz2  11966  or3di  29698  ifpan23  38412  ifpidg  38444  ifpim123g  38453
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