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Theorem in32 3467
 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 3465 . 2 ((AB) ∩ C) = (A ∩ (BC))
2 in12 3466 . 2 (A ∩ (BC)) = (B ∩ (AC))
3 incom 3448 . 2 (B ∩ (AC)) = ((AC) ∩ B)
41, 2, 33eqtri 2377 1 ((AB) ∩ C) = ((AC) ∩ B)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642   ∩ cin 3208 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213 This theorem is referenced by:  in13  3468  inrot  3470
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