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Theorem inindir 3262
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 3253 . . 3 (𝐶𝐶) = 𝐶
21ineq2i 3242 . 2 ((𝐴𝐵) ∩ (𝐶𝐶)) = ((𝐴𝐵) ∩ 𝐶)
3 in4 3260 . 2 ((𝐴𝐵) ∩ (𝐶𝐶)) = ((𝐴𝐶) ∩ (𝐵𝐶))
42, 3eqtr3i 2138 1 ((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:   = wceq 1314  cin 3038
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 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 2245  df-v 2660  df-in 3045
This theorem is referenced by:  difindir  3299  resindir  4803  restbasg  12232
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