Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  inindi GIF version

Theorem inindi 3259
 Description: Intersection distributes over itself. (Contributed by NM, 6-May-1994.)
Assertion
Ref Expression
inindi (𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))

Proof of Theorem inindi
StepHypRef Expression
1 inidm 3251 . . 3 (𝐴𝐴) = 𝐴
21ineq1i 3239 . 2 ((𝐴𝐴) ∩ (𝐵𝐶)) = (𝐴 ∩ (𝐵𝐶))
3 in4 3258 . 2 ((𝐴𝐴) ∩ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))
42, 3eqtr3i 2137 1 (𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))
 Colors of variables: wff set class Syntax hints:   = wceq 1314   ∩ cin 3036 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097 This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-in 3043 This theorem is referenced by:  resindi  4792  offres  5987
 Copyright terms: Public domain W3C validator