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Theorem indi 4274
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 1007 . . . 4 ((𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∨ (𝑥𝐴𝑥𝐶)))
2 elin 3965 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
3 elin 3965 . . . . 5 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴𝑥𝐶))
42, 3orbi12i 914 . . . 4 ((𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐴𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∨ (𝑥𝐴𝑥𝐶)))
51, 4bitr4i 278 . . 3 ((𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)) ↔ (𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐴𝐶)))
6 elun 4149 . . . 4 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
76anbi2i 624 . . 3 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)))
8 elun 4149 . . 3 (𝑥 ∈ ((𝐴𝐵) ∪ (𝐴𝐶)) ↔ (𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐴𝐶)))
95, 7, 83bitr4i 303 . 2 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ 𝑥 ∈ ((𝐴𝐵) ∪ (𝐴𝐶)))
109ineqri 4205 1 (𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  wa 397  wo 846   = wceq 1542  wcel 2107  cun 3947  cin 3948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-un 3954  df-in 3956
This theorem is referenced by:  indir  4276  difindi  4282  undisj2  4463  disjssun  4468  difdifdir  4492  disjpr2  4718  resundi  5996  fresaun  6763  elfiun  9425  unxpwdom  9584  kmlem2  10146  djuinf  10183  ackbij1lem1  10215  ackbij1lem2  10216  ssxr  11283  incexclem  15782  bitsinv1  16383  bitsinvp1  16390  bitsres  16414  paste  22798  unmbl  25054  ovolioo  25085  uniioombllem4  25103  volcn  25123  ellimc2  25394  lhop2  25532  ex-in  29678  eulerpartgbij  33371  poimirlem3  36491  poimirlem15  36503  asindmre  36571  iunrelexp0  42453  sge0resplit  45122  sge0split  45125  iscnrm3rlem1  47573  topdlat  47629
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