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Theorem ordi 740
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 106 . . . 4 ((𝜓𝜒) → 𝜓)
21orim2i 688 . . 3 ((𝜑 ∨ (𝜓𝜒)) → (𝜑𝜓))
3 simpr 107 . . . 4 ((𝜓𝜒) → 𝜒)
43orim2i 688 . . 3 ((𝜑 ∨ (𝜓𝜒)) → (𝜑𝜒))
52, 4jca 294 . 2 ((𝜑 ∨ (𝜓𝜒)) → ((𝜑𝜓) ∧ (𝜑𝜒)))
6 orc 643 . . . 4 (𝜑 → (𝜑 ∨ (𝜓𝜒)))
76adantl 266 . . 3 (((𝜑𝜓) ∧ 𝜑) → (𝜑 ∨ (𝜓𝜒)))
86adantr 265 . . . 4 ((𝜑𝜒) → (𝜑 ∨ (𝜓𝜒)))
9 olc 642 . . . 4 ((𝜓𝜒) → (𝜑 ∨ (𝜓𝜒)))
108, 9jaoian 719 . . 3 (((𝜑𝜓) ∧ 𝜒) → (𝜑 ∨ (𝜓𝜒)))
117, 10jaodan 721 . 2 (((𝜑𝜓) ∧ (𝜑𝜒)) → (𝜑 ∨ (𝜓𝜒)))
125, 11impbii 121 1 ((𝜑 ∨ (𝜓𝜒)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102  wo 639
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  ordir  741  orddi  744  pm5.63dc  864  pm4.43  867  orbididc  871  undi  3213  undif4  3312  elnn1uz2  8641
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