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Theorem in32 4220
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 4218 . 2 ((𝐴𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵𝐶))
2 in12 4219 . 2 (𝐴 ∩ (𝐵𝐶)) = (𝐵 ∩ (𝐴𝐶))
3 incom 4199 . 2 (𝐵 ∩ (𝐴𝐶)) = ((𝐴𝐶) ∩ 𝐵)
41, 2, 33eqtri 2757 1 ((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∩ 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  cin 3943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3419  df-v 3463  df-in 3951
This theorem is referenced by:  in13  4221  inrot  4223  wefrc  5672  imainrect  6187  sspred  6316  fpwwe2  10673  incexclem  15826  setsfun  17159  setsfun0  17160  ressress  17248  kgeni  23502  kgencn3  23523  fclsrest  23989  voliunlem1  25540  bj-disj2r  36658  refrelsredund4  38254
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