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Theorem indi 3501
 Description: Distributive law for intersection over union. Exercise 10 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
indi (A ∩ (BC)) = ((AB) ∪ (AC))

Proof of Theorem indi
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 andi 837 . . . 4 ((x A (x B x C)) ↔ ((x A x B) (x A x C)))
2 elin 3219 . . . . 5 (x (AB) ↔ (x A x B))
3 elin 3219 . . . . 5 (x (AC) ↔ (x A x C))
42, 3orbi12i 507 . . . 4 ((x (AB) x (AC)) ↔ ((x A x B) (x A x C)))
51, 4bitr4i 243 . . 3 ((x A (x B x C)) ↔ (x (AB) x (AC)))
6 elun 3220 . . . 4 (x (BC) ↔ (x B x C))
76anbi2i 675 . . 3 ((x A x (BC)) ↔ (x A (x B x C)))
8 elun 3220 . . 3 (x ((AB) ∪ (AC)) ↔ (x (AB) x (AC)))
95, 7, 83bitr4i 268 . 2 ((x A x (BC)) ↔ x ((AB) ∪ (AC)))
109ineqri 3449 1 (A ∩ (BC)) = ((AB) ∪ (AC))
 Colors of variables: wff setvar class Syntax hints:   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ∪ cun 3207   ∩ cin 3208 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214 This theorem is referenced by:  indir  3503  difindi  3509  undisj2  3603  disjssun  3608  difdifdir  3637  diftpsn3  3849  addcass  4415  resundi  4981  sbthlem1  6203
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