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Theorem undir 4019
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 4017 . 2 (𝐶 ∪ (𝐴𝐵)) = ((𝐶𝐴) ∩ (𝐶𝐵))
2 uncom 3900 . 2 ((𝐴𝐵) ∪ 𝐶) = (𝐶 ∪ (𝐴𝐵))
3 uncom 3900 . . 3 (𝐴𝐶) = (𝐶𝐴)
4 uncom 3900 . . 3 (𝐵𝐶) = (𝐶𝐵)
53, 4ineq12i 3955 . 2 ((𝐴𝐶) ∩ (𝐵𝐶)) = ((𝐶𝐴) ∩ (𝐶𝐵))
61, 2, 53eqtr4i 2792 1 ((𝐴𝐵) ∪ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1632  cun 3713  cin 3714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-un 3720  df-in 3722
This theorem is referenced by:  undif1  4187  dfif4  4245  dfif5  4246  bwth  21435
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