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Theorem undi 3398
Description: Distributive law for union over intersection. Exercise 11 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
undi (𝐴 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))

Proof of Theorem undi
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elin 3333 . . . 4 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
21orbi2i 763 . . 3 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 ∨ (𝑥𝐵𝑥𝐶)))
3 ordi 817 . . 3 ((𝑥𝐴 ∨ (𝑥𝐵𝑥𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴𝑥𝐶)))
4 elin 3333 . . . 4 (𝑥 ∈ ((𝐴𝐵) ∩ (𝐴𝐶)) ↔ (𝑥 ∈ (𝐴𝐵) ∧ 𝑥 ∈ (𝐴𝐶)))
5 elun 3291 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
6 elun 3291 . . . . 5 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴𝑥𝐶))
75, 6anbi12i 460 . . . 4 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑥 ∈ (𝐴𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴𝑥𝐶)))
84, 7bitr2i 185 . . 3 (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴𝑥𝐶)) ↔ 𝑥 ∈ ((𝐴𝐵) ∩ (𝐴𝐶)))
92, 3, 83bitri 206 . 2 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ 𝑥 ∈ ((𝐴𝐵) ∩ (𝐴𝐶)))
109uneqri 3292 1 (𝐴 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))
Colors of variables: wff set class
Syntax hints:  wa 104  wo 709   = wceq 1364  wcel 2160  cun 3142  cin 3143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-in 3150
This theorem is referenced by:  undir  3400  undifdc  6952
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