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Theorem in12 3973
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 3956 . . 3 (𝐴𝐵) = (𝐵𝐴)
21ineq1i 3961 . 2 ((𝐴𝐵) ∩ 𝐶) = ((𝐵𝐴) ∩ 𝐶)
3 inass 3972 . 2 ((𝐴𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵𝐶))
4 inass 3972 . 2 ((𝐵𝐴) ∩ 𝐶) = (𝐵 ∩ (𝐴𝐶))
52, 3, 43eqtr3i 2801 1 (𝐴 ∩ (𝐵𝐶)) = (𝐵 ∩ (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  cin 3722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-in 3730
This theorem is referenced by:  in32  3974  in31  3976  in4  3978  resdmres  5768  kmlem12  9189  ressress  16146  fh1  28817  fh2  28818  mdslmd3i  29531  bj-inrab3  33257
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