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Theorem ordi 926
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 925 . 2 ((¬ 𝜑 → (𝜓𝜒)) ↔ ((¬ 𝜑𝜓) ∧ (¬ 𝜑𝜒)))
2 df-or 384 . 2 ((𝜑 ∨ (𝜓𝜒)) ↔ (¬ 𝜑 → (𝜓𝜒)))
3 df-or 384 . . 3 ((𝜑𝜓) ↔ (¬ 𝜑𝜓))
4 df-or 384 . . 3 ((𝜑𝜒) ↔ (¬ 𝜑𝜒))
53, 4anbi12i 733 . 2 (((𝜑𝜓) ∧ (𝜑𝜒)) ↔ ((¬ 𝜑𝜓) ∧ (¬ 𝜑𝜒)))
61, 2, 53bitr4i 292 1 ((𝜑 ∨ (𝜓𝜒)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383
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
This theorem is referenced by:  ordir  927  orddi  931  pm5.63  979  pm4.43  988  cadan  1588  undi  3907  undif3  3921  undif4  4068  elnn1uz2  11803  or3di  29435  ifpan23  38121  ifpidg  38153  ifpim123g  38162
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