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Theorem ordir 1004
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 1003 . 2 ((𝜒 ∨ (𝜑𝜓)) ↔ ((𝜒𝜑) ∧ (𝜒𝜓)))
2 orcom 867 . 2 (((𝜑𝜓) ∨ 𝜒) ↔ (𝜒 ∨ (𝜑𝜓)))
3 orcom 867 . . 3 ((𝜑𝜒) ↔ (𝜒𝜑))
4 orcom 867 . . 3 ((𝜓𝜒) ↔ (𝜒𝜓))
53, 4anbi12i 629 . 2 (((𝜑𝜒) ∧ (𝜓𝜒)) ↔ ((𝜒𝜑) ∧ (𝜒𝜓)))
61, 2, 53bitr4i 306 1 (((𝜑𝜓) ∨ 𝜒) ↔ ((𝜑𝜒) ∧ (𝜓𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wo 844
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 210  df-an 400  df-or 845
This theorem is referenced by:  orddi  1007  pm5.62  1016  dn1  1053  cadan  1611  elnn0z  11986  ifpim123g  40195  rp-fakeanorass  40208  fvmptrabdm  43836
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