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Theorem ordi 788
Description: Distributive law for disjunction. Theorem *4.41 of [WhiteheadRussell] p. 119. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
Assertion
Ref Expression
ordi ((𝜑 ∨ (𝜓𝜒)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))

Proof of Theorem ordi
StepHypRef Expression
1 simpl 108 . . . 4 ((𝜓𝜒) → 𝜓)
21orim2i 733 . . 3 ((𝜑 ∨ (𝜓𝜒)) → (𝜑𝜓))
3 simpr 109 . . . 4 ((𝜓𝜒) → 𝜒)
43orim2i 733 . . 3 ((𝜑 ∨ (𝜓𝜒)) → (𝜑𝜒))
52, 4jca 302 . 2 ((𝜑 ∨ (𝜓𝜒)) → ((𝜑𝜓) ∧ (𝜑𝜒)))
6 orc 684 . . . 4 (𝜑 → (𝜑 ∨ (𝜓𝜒)))
76adantl 273 . . 3 (((𝜑𝜓) ∧ 𝜑) → (𝜑 ∨ (𝜓𝜒)))
86adantr 272 . . . 4 ((𝜑𝜒) → (𝜑 ∨ (𝜓𝜒)))
9 olc 683 . . . 4 ((𝜓𝜒) → (𝜑 ∨ (𝜓𝜒)))
108, 9jaoian 767 . . 3 (((𝜑𝜓) ∧ 𝜒) → (𝜑 ∨ (𝜓𝜒)))
117, 10jaodan 769 . 2 (((𝜑𝜓) ∧ (𝜑𝜒)) → (𝜑 ∨ (𝜓𝜒)))
125, 11impbii 125 1 ((𝜑 ∨ (𝜓𝜒)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wo 680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  ordir  789  orddi  792  pm5.63dc  913  pm4.43  916  orbididc  920  undi  3292  undif4  3393  elnn1uz2  9353
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