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Theorem andi 742
Description: Distributive law for conjunction. Theorem *4.4 of [WhiteheadRussell] p. 118. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 5-Jan-2013.)
Assertion
Ref Expression
andi ((𝜑 ∧ (𝜓𝜒)) ↔ ((𝜑𝜓) ∨ (𝜑𝜒)))

Proof of Theorem andi
StepHypRef Expression
1 orc 643 . . 3 ((𝜑𝜓) → ((𝜑𝜓) ∨ (𝜑𝜒)))
2 olc 642 . . 3 ((𝜑𝜒) → ((𝜑𝜓) ∨ (𝜑𝜒)))
31, 2jaodan 721 . 2 ((𝜑 ∧ (𝜓𝜒)) → ((𝜑𝜓) ∨ (𝜑𝜒)))
4 orc 643 . . . 4 (𝜓 → (𝜓𝜒))
54anim2i 328 . . 3 ((𝜑𝜓) → (𝜑 ∧ (𝜓𝜒)))
6 olc 642 . . . 4 (𝜒 → (𝜓𝜒))
76anim2i 328 . . 3 ((𝜑𝜒) → (𝜑 ∧ (𝜓𝜒)))
85, 7jaoi 646 . 2 (((𝜑𝜓) ∨ (𝜑𝜒)) → (𝜑 ∧ (𝜓𝜒)))
93, 8impbii 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:  andir  743  anddi  745  dcim  795  dcan  853  excxor  1285  sbequilem  1735  sborv  1786  r19.43  2485  indi  3212  difindiss  3219  unrab  3236  unipr  3622  uniun  3627  unopab  3864  xpundi  4424  coundir  4851  unpreima  5320  tpostpos  5910  elni2  6470  elznn0nn  8316
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