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Theorem in12 3802
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 3783 . . 3 (𝐴𝐵) = (𝐵𝐴)
21ineq1i 3788 . 2 ((𝐴𝐵) ∩ 𝐶) = ((𝐵𝐴) ∩ 𝐶)
3 inass 3801 . 2 ((𝐴𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵𝐶))
4 inass 3801 . 2 ((𝐵𝐴) ∩ 𝐶) = (𝐵 ∩ (𝐴𝐶))
52, 3, 43eqtr3i 2651 1 (𝐴 ∩ (𝐵𝐶)) = (𝐵 ∩ (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  cin 3554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-in 3562
This theorem is referenced by:  in32  3803  in31  3805  in4  3807  resdmres  5584  kmlem12  8927  ressress  15859  fh1  28323  fh2  28324  mdslmd3i  29037  bj-inrab3  32569
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