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Theorem in12 3211
Description: A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.)
Assertion
Ref Expression
in12 (𝐴 ∩ (𝐵𝐶)) = (𝐵 ∩ (𝐴𝐶))

Proof of Theorem in12
StepHypRef Expression
1 incom 3192 . . 3 (𝐴𝐵) = (𝐵𝐴)
21ineq1i 3197 . 2 ((𝐴𝐵) ∩ 𝐶) = ((𝐵𝐴) ∩ 𝐶)
3 inass 3210 . 2 ((𝐴𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵𝐶))
4 inass 3210 . 2 ((𝐵𝐴) ∩ 𝐶) = (𝐵 ∩ (𝐴𝐶))
52, 3, 43eqtr3i 2116 1 (𝐴 ∩ (𝐵𝐶)) = (𝐵 ∩ (𝐴𝐶))
Colors of variables: wff set class
Syntax hints:   = wceq 1289  cin 2998
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3005
This theorem is referenced by:  in32  3212  in31  3214  in4  3216  resdmres  4922
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