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Theorem in32 4161
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 4159 . 2 ((𝐴𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵𝐶))
2 in12 4160 . 2 (𝐴 ∩ (𝐵𝐶)) = (𝐵 ∩ (𝐴𝐶))
3 incom 4140 . 2 (𝐵 ∩ (𝐴𝐶)) = ((𝐴𝐶) ∩ 𝐵)
41, 2, 33eqtri 2772 1 ((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∩ 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  cin 3891
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 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1545  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-rab 3075  df-v 3433  df-in 3899
This theorem is referenced by:  in13  4162  inrot  4164  wefrc  5583  imainrect  6082  sspred  6209  fpwwe2  10392  incexclem  15538  setsfun  16862  setsfun0  16863  ressress  16948  kgeni  22678  kgencn3  22699  fclsrest  23165  voliunlem1  24704  bj-disj2r  35206  refrelsredund4  36733
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