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Theorem indi 4200
 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 1005 . . . 4 ((𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∨ (𝑥𝐴𝑥𝐶)))
2 elin 3897 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
3 elin 3897 . . . . 5 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴𝑥𝐶))
42, 3orbi12i 912 . . . 4 ((𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐴𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∨ (𝑥𝐴𝑥𝐶)))
51, 4bitr4i 281 . . 3 ((𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)) ↔ (𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐴𝐶)))
6 elun 4076 . . . 4 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
76anbi2i 625 . . 3 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)))
8 elun 4076 . . 3 (𝑥 ∈ ((𝐴𝐵) ∪ (𝐴𝐶)) ↔ (𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐴𝐶)))
95, 7, 83bitr4i 306 . 2 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ 𝑥 ∈ ((𝐴𝐵) ∪ (𝐴𝐶)))
109ineqri 4130 1 (𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111   ∪ cun 3879   ∩ cin 3880 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-un 3886  df-in 3888 This theorem is referenced by:  indir  4202  difindi  4208  undisj2  4370  disjssun  4375  difdifdir  4395  disjpr2  4609  resundi  5833  fresaun  6524  elfiun  8881  unxpwdom  9040  kmlem2  9565  djuinf  9602  ackbij1lem1  9634  ackbij1lem2  9635  ssxr  10702  incexclem  15186  bitsinv1  15784  bitsinvp1  15791  bitsres  15815  paste  21909  unmbl  24151  ovolioo  24182  uniioombllem4  24200  volcn  24220  ellimc2  24490  lhop2  24628  ex-in  28220  eulerpartgbij  31755  poimirlem3  35079  poimirlem15  35091  asindmre  35159  iunrelexp0  40446  sge0resplit  43088  sge0split  43091
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