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Theorem ordir 817
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 816 . 2 ((𝜒 ∨ (𝜑𝜓)) ↔ ((𝜒𝜑) ∧ (𝜒𝜓)))
2 orcom 728 . 2 (((𝜑𝜓) ∨ 𝜒) ↔ (𝜒 ∨ (𝜑𝜓)))
3 orcom 728 . . 3 ((𝜑𝜒) ↔ (𝜒𝜑))
4 orcom 728 . . 3 ((𝜓𝜒) ↔ (𝜒𝜓))
53, 4anbi12i 460 . 2 (((𝜑𝜒) ∧ (𝜓𝜒)) ↔ ((𝜒𝜑) ∧ (𝜒𝜓)))
61, 2, 53bitr4i 212 1 (((𝜑𝜓) ∨ 𝜒) ↔ ((𝜑𝜒) ∧ (𝜓𝜒)))
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105  wo 708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709
This theorem depends on definitions:  df-bi 117
This theorem is referenced by:  orddi  820  pm5.62dc  945  dn1dc  960  suc11g  4552  bj-peano4  14329
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