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Theorem inindir 3325
 Description: Intersection distributes over itself. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
inindir ((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))

Proof of Theorem inindir
StepHypRef Expression
1 inidm 3316 . . 3 (𝐶𝐶) = 𝐶
21ineq2i 3305 . 2 ((𝐴𝐵) ∩ (𝐶𝐶)) = ((𝐴𝐵) ∩ 𝐶)
3 in4 3323 . 2 ((𝐴𝐵) ∩ (𝐶𝐶)) = ((𝐴𝐶) ∩ (𝐵𝐶))
42, 3eqtr3i 2180 1 ((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))
 Colors of variables: wff set class Syntax hints:   = wceq 1335   ∩ cin 3101 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-in 3108 This theorem is referenced by:  difindir  3362  resindir  4879  restbasg  12528
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