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Theorem undir 3265
 Description: Distributive law for union over intersection. Theorem 29 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)
Assertion
Ref Expression
undir ((𝐴𝐵) ∪ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))

Proof of Theorem undir
StepHypRef Expression
1 undi 3263 . 2 (𝐶 ∪ (𝐴𝐵)) = ((𝐶𝐴) ∩ (𝐶𝐵))
2 uncom 3159 . 2 ((𝐴𝐵) ∪ 𝐶) = (𝐶 ∪ (𝐴𝐵))
3 uncom 3159 . . 3 (𝐴𝐶) = (𝐶𝐴)
4 uncom 3159 . . 3 (𝐵𝐶) = (𝐶𝐵)
53, 4ineq12i 3214 . 2 ((𝐴𝐶) ∩ (𝐵𝐶)) = ((𝐶𝐴) ∩ (𝐶𝐵))
61, 2, 53eqtr4i 2125 1 ((𝐴𝐵) ∪ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))
 Colors of variables: wff set class Syntax hints:   = wceq 1296   ∪ cun 3011   ∩ cin 3012 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077 This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-un 3017  df-in 3019 This theorem is referenced by: (None)
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