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Theorem in12 3466
 Description: A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.)
Assertion
Ref Expression
in12 (A ∩ (BC)) = (B ∩ (AC))

Proof of Theorem in12
StepHypRef Expression
1 incom 3448 . . 3 (AB) = (BA)
21ineq1i 3453 . 2 ((AB) ∩ C) = ((BA) ∩ C)
3 inass 3465 . 2 ((AB) ∩ C) = (A ∩ (BC))
4 inass 3465 . 2 ((BA) ∩ C) = (B ∩ (AC))
52, 3, 43eqtr3i 2381 1 (A ∩ (BC)) = (B ∩ (AC))
 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:  in32  3467  in31  3469  in4  3471  resdmres  5078
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