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Theorem in32 3213
 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 3211 . 2 ((𝐴𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵𝐶))
2 in12 3212 . 2 (𝐴 ∩ (𝐵𝐶)) = (𝐵 ∩ (𝐴𝐶))
3 incom 3193 . 2 (𝐵 ∩ (𝐴𝐶)) = ((𝐴𝐶) ∩ 𝐵)
41, 2, 33eqtri 2113 1 ((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∩ 𝐵)
 Colors of variables: wff set class Syntax hints:   = wceq 1290   ∩ cin 2999 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071 This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-in 3006 This theorem is referenced by:  in13  3214  inrot  3216  imainrect  4889  setsfun  11583  setsfun0  11584
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