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Theorem indir 3398
Description: Distributive law for intersection over union. Theorem 28 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)
Assertion
Ref Expression
indir ((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))

Proof of Theorem indir
StepHypRef Expression
1 indi 3396 . 2 (𝐶 ∩ (𝐴𝐵)) = ((𝐶𝐴) ∪ (𝐶𝐵))
2 incom 3341 . 2 ((𝐴𝐵) ∩ 𝐶) = (𝐶 ∩ (𝐴𝐵))
3 incom 3341 . . 3 (𝐴𝐶) = (𝐶𝐴)
4 incom 3341 . . 3 (𝐵𝐶) = (𝐶𝐵)
53, 4uneq12i 3301 . 2 ((𝐴𝐶) ∪ (𝐵𝐶)) = ((𝐶𝐴) ∪ (𝐶𝐵))
61, 2, 53eqtr4i 2219 1 ((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:   = wceq 1363  cun 3141  cin 3142
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2170
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-v 2753  df-un 3147  df-in 3149
This theorem is referenced by:  difundir  3402  undisj1  3494  disjpr2  3670  resundir  4935  djuassen  7233
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