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Theorem in32 3787
 Description: A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
in32 ((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∩ 𝐵)

Proof of Theorem in32
StepHypRef Expression
1 inass 3785 . 2 ((𝐴𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵𝐶))
2 in12 3786 . 2 (𝐴 ∩ (𝐵𝐶)) = (𝐵 ∩ (𝐴𝐶))
3 incom 3767 . 2 (𝐵 ∩ (𝐴𝐶)) = ((𝐴𝐶) ∩ 𝐵)
41, 2, 33eqtri 2636 1 ((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∩ 𝐵)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∩ cin 3539 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547 This theorem is referenced by:  in13  3788  inrot  3790  wefrc  5032  imainrect  5494  sspred  5605  fpwwe2  9344  incexclem  14407  setsfun  15725  setsfun0  15726  ressress  15765  kgeni  21150  kgencn3  21171  fclsrest  21638  voliunlem1  23125
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