New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  indir GIF version

Theorem indir 3503
 Description: Distributive law for intersection over union. Theorem 28 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)
Assertion
Ref Expression
indir ((AB) ∩ C) = ((AC) ∪ (BC))

Proof of Theorem indir
StepHypRef Expression
1 indi 3501 . 2 (C ∩ (AB)) = ((CA) ∪ (CB))
2 incom 3448 . 2 ((AB) ∩ C) = (C ∩ (AB))
3 incom 3448 . . 3 (AC) = (CA)
4 incom 3448 . . 3 (BC) = (CB)
53, 4uneq12i 3416 . 2 ((AC) ∪ (BC)) = ((CA) ∪ (CB))
61, 2, 53eqtr4i 2383 1 ((AB) ∩ C) = ((AC) ∪ (BC))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642   ∪ cun 3207   ∩ cin 3208 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214 This theorem is referenced by:  difundir  3508  undisj1  3602  addcass  4415  nnsucelrlem3  4426  resundir  4982
 Copyright terms: Public domain W3C validator