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Theorem in31 3789
 Description: A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)
Assertion
Ref Expression
in31 ((𝐴𝐵) ∩ 𝐶) = ((𝐶𝐵) ∩ 𝐴)

Proof of Theorem in31
StepHypRef Expression
1 in12 3786 . 2 (𝐶 ∩ (𝐴𝐵)) = (𝐴 ∩ (𝐶𝐵))
2 incom 3767 . 2 ((𝐴𝐵) ∩ 𝐶) = (𝐶 ∩ (𝐴𝐵))
3 incom 3767 . 2 ((𝐶𝐵) ∩ 𝐴) = (𝐴 ∩ (𝐶𝐵))
41, 2, 33eqtr4i 2642 1 ((𝐴𝐵) ∩ 𝐶) = ((𝐶𝐵) ∩ 𝐴)
 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:  inrot  3790
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