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Theorem ordi 1000
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 518 . 2 ((¬ 𝜑 → (𝜓𝜒)) ↔ ((¬ 𝜑𝜓) ∧ (¬ 𝜑𝜒)))
2 df-or 843 . 2 ((𝜑 ∨ (𝜓𝜒)) ↔ (¬ 𝜑 → (𝜓𝜒)))
3 df-or 843 . . 3 ((𝜑𝜓) ↔ (¬ 𝜑𝜓))
4 df-or 843 . . 3 ((𝜑𝜒) ↔ (¬ 𝜑𝜒))
53, 4anbi12i 626 . 2 (((𝜑𝜓) ∧ (𝜑𝜒)) ↔ ((¬ 𝜑𝜓) ∧ (¬ 𝜑𝜒)))
61, 2, 53bitr4i 304 1 ((𝜑 ∨ (𝜓𝜒)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 842
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 843
This theorem is referenced by:  ordir  1001  orddi  1004  pm5.63  1014  pm4.43  1017  cadan  1591  undi  4171  undif3  4185  undif4  4330  elnn1uz2  12174  or3di  29916  ifpan23  39329  ifpidg  39361  ifpim123g  39370
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