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Theorem ordir 905
 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 904 . 2 ((𝜒 ∨ (𝜑𝜓)) ↔ ((𝜒𝜑) ∧ (𝜒𝜓)))
2 orcom 401 . 2 (((𝜑𝜓) ∨ 𝜒) ↔ (𝜒 ∨ (𝜑𝜓)))
3 orcom 401 . . 3 ((𝜑𝜒) ↔ (𝜒𝜑))
4 orcom 401 . . 3 ((𝜓𝜒) ↔ (𝜒𝜓))
53, 4anbi12i 729 . 2 (((𝜑𝜒) ∧ (𝜓𝜒)) ↔ ((𝜒𝜑) ∧ (𝜒𝜓)))
61, 2, 53bitr4i 291 1 (((𝜑𝜓) ∨ 𝜒) ↔ ((𝜑𝜒) ∧ (𝜓𝜒)))
 Colors of variables: wff setvar class Syntax hints:   ↔ 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:  orddi  909  pm5.62  960  dn1  1000  cadan  1539  elnn0z  11267  ifpim123g  36864  rp-fakeanorass  36877
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