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

Theorem ordir 1000
Description: Distributive law for disjunction. (Contributed by NM, 12-Aug-1994.)
Assertion
Ref Expression
ordir (((𝜑𝜓) ∨ 𝜒) ↔ ((𝜑𝜒) ∧ (𝜓𝜒)))

Proof of Theorem ordir
StepHypRef Expression
1 ordi 999 . 2 ((𝜒 ∨ (𝜑𝜓)) ↔ ((𝜒𝜑) ∧ (𝜒𝜓)))
2 orcom 864 . 2 (((𝜑𝜓) ∨ 𝜒) ↔ (𝜒 ∨ (𝜑𝜓)))
3 orcom 864 . . 3 ((𝜑𝜒) ↔ (𝜒𝜑))
4 orcom 864 . . 3 ((𝜓𝜒) ↔ (𝜒𝜓))
53, 4anbi12i 626 . 2 (((𝜑𝜒) ∧ (𝜓𝜒)) ↔ ((𝜒𝜑) ∧ (𝜒𝜓)))
61, 2, 53bitr4i 304 1 (((𝜑𝜓) ∨ 𝜒) ↔ ((𝜑𝜒) ∧ (𝜓𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wo 841
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 842
This theorem is referenced by:  orddi  1003  pm5.62  1012  dn1  1049  cadan  1601  elnn0z  11982  ifpim123g  39744  rp-fakeanorass  39757  fvmptrabdm  43369
  Copyright terms: Public domain W3C validator