Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  in31 Structured version   Visualization version   GIF version

Theorem in31 3976
 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 3973 . 2 (𝐶 ∩ (𝐴𝐵)) = (𝐴 ∩ (𝐶𝐵))
2 incom 3956 . 2 ((𝐴𝐵) ∩ 𝐶) = (𝐶 ∩ (𝐴𝐵))
3 incom 3956 . 2 ((𝐶𝐵) ∩ 𝐴) = (𝐴 ∩ (𝐶𝐵))
41, 2, 33eqtr4i 2803 1 ((𝐴𝐵) ∩ 𝐶) = ((𝐶𝐵) ∩ 𝐴)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1631   ∩ cin 3722 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-in 3730 This theorem is referenced by:  inrot  3977
 Copyright terms: Public domain W3C validator