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Theorem in12 4112
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 4092 . . 3 (𝐴𝐵) = (𝐵𝐴)
21ineq1i 4100 . 2 ((𝐴𝐵) ∩ 𝐶) = ((𝐵𝐴) ∩ 𝐶)
3 inass 4111 . 2 ((𝐴𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵𝐶))
4 inass 4111 . 2 ((𝐵𝐴) ∩ 𝐶) = (𝐵 ∩ (𝐴𝐶))
52, 3, 43eqtr3i 2769 1 (𝐴 ∩ (𝐵𝐶)) = (𝐵 ∩ (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  cin 3843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-rab 3062  df-v 3400  df-in 3851
This theorem is referenced by:  in32  4113  in31  4115  in4  4117  resdmres  6065  kmlem12  9662  ressress  16666  fh1  29553  fh2  29554  mdslmd3i  30267  bj-inrab3  34745
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