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Theorem undir 4041
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 4039 . 2 (𝐶 ∪ (𝐴𝐵)) = ((𝐶𝐴) ∩ (𝐶𝐵))
2 uncom 3919 . 2 ((𝐴𝐵) ∪ 𝐶) = (𝐶 ∪ (𝐴𝐵))
3 uncom 3919 . . 3 (𝐴𝐶) = (𝐶𝐴)
4 uncom 3919 . . 3 (𝐵𝐶) = (𝐶𝐵)
53, 4ineq12i 3974 . 2 ((𝐴𝐶) ∩ (𝐵𝐶)) = ((𝐶𝐴) ∩ (𝐶𝐵))
61, 2, 53eqtr4i 2797 1 ((𝐴𝐵) ∪ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1652  cun 3730  cin 3731
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352  df-un 3737  df-in 3739
This theorem is referenced by:  undif1  4203  dfif4  4258  dfif5  4259  bwth  21493
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