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Theorem undir 3835
 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 3833 . 2 (𝐶 ∪ (𝐴𝐵)) = ((𝐶𝐴) ∩ (𝐶𝐵))
2 uncom 3719 . 2 ((𝐴𝐵) ∪ 𝐶) = (𝐶 ∪ (𝐴𝐵))
3 uncom 3719 . . 3 (𝐴𝐶) = (𝐶𝐴)
4 uncom 3719 . . 3 (𝐵𝐶) = (𝐶𝐵)
53, 4ineq12i 3774 . 2 ((𝐴𝐶) ∩ (𝐵𝐶)) = ((𝐶𝐴) ∩ (𝐶𝐵))
61, 2, 53eqtr4i 2642 1 ((𝐴𝐵) ∪ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∪ cun 3538   ∩ cin 3539 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-un 3545  df-in 3547 This theorem is referenced by:  undif1  3995  dfif4  4051  dfif5  4052  bwth  21023
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