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Theorem undir 4237
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 4235 . 2 (𝐶 ∪ (𝐴𝐵)) = ((𝐶𝐴) ∩ (𝐶𝐵))
2 uncom 4114 . 2 ((𝐴𝐵) ∪ 𝐶) = (𝐶 ∪ (𝐴𝐵))
3 uncom 4114 . . 3 (𝐴𝐶) = (𝐶𝐴)
4 uncom 4114 . . 3 (𝐵𝐶) = (𝐶𝐵)
53, 4ineq12i 4171 . 2 ((𝐴𝐶) ∩ (𝐵𝐶)) = ((𝐶𝐴) ∩ (𝐶𝐵))
61, 2, 53eqtr4i 2857 1 ((𝐴𝐵) ∪ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  cun 3917  cin 3918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-rab 3142  df-v 3482  df-un 3924  df-in 3926
This theorem is referenced by:  undif1  4406  dfif4  4464  dfif5  4465  bwth  22011
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