MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  in32 Structured version   Visualization version   GIF version

Theorem in32 3974
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 3972 . 2 ((𝐴𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵𝐶))
2 in12 3973 . 2 (𝐴 ∩ (𝐵𝐶)) = (𝐵 ∩ (𝐴𝐶))
3 incom 3956 . 2 (𝐵 ∩ (𝐴𝐶)) = ((𝐴𝐶) ∩ 𝐵)
41, 2, 33eqtri 2797 1 ((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∩ 𝐵)
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 835  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:  in13  3975  inrot  3977  wefrc  5243  imainrect  5716  sspred  5831  fpwwe2  9667  incexclem  14775  setsfun  16100  setsfun0  16101  ressress  16146  kgeni  21561  kgencn3  21582  fclsrest  22048  voliunlem1  23538  bj-disj2r  33344
  Copyright terms: Public domain W3C validator