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Theorem ordi 990
 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 507 . 2 ((¬ 𝜑 → (𝜓𝜒)) ↔ ((¬ 𝜑𝜓) ∧ (¬ 𝜑𝜒)))
2 df-or 837 . 2 ((𝜑 ∨ (𝜓𝜒)) ↔ (¬ 𝜑 → (𝜓𝜒)))
3 df-or 837 . . 3 ((𝜑𝜓) ↔ (¬ 𝜑𝜓))
4 df-or 837 . . 3 ((𝜑𝜒) ↔ (¬ 𝜑𝜒))
53, 4anbi12i 612 . 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   ∧ wa 382   ∨ wo 836 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-an 383  df-or 837 This theorem is referenced by:  ordir  991  orddi  994  pm5.63  1005  pm4.43  1008  cadan  1696  undi  4023  undif3  4037  undif4  4178  elnn1uz2  11973  or3di  29647  ifpan23  38330  ifpidg  38362  ifpim123g  38371
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