Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ordi Structured version   Visualization version   GIF version

Theorem ordi 904
 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 903 . 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 729 . 2 (((𝜑𝜓) ∧ (𝜑𝜒)) ↔ ((¬ 𝜑𝜓) ∧ (¬ 𝜑𝜒)))
61, 2, 53bitr4i 291 1 ((𝜑 ∨ (𝜓𝜒)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ 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 196  df-or 384  df-an 385 This theorem is referenced by:  ordir  905  orddi  909  pm5.63  961  pm4.43  964  cadan  1539  undi  3833  undif3  3847  undif4  3987  elnn1uz2  11641  or3di  28691  ifpan23  36823  ifpidg  36855  ifpim123g  36864
 Copyright terms: Public domain W3C validator