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Theorem inindir 3473
 Description: Intersection distributes over itself. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
inindir ((AB) ∩ C) = ((AC) ∩ (BC))

Proof of Theorem inindir
StepHypRef Expression
1 inidm 3464 . . 3 (CC) = C
21ineq2i 3454 . 2 ((AB) ∩ (CC)) = ((AB) ∩ C)
3 in4 3471 . 2 ((AB) ∩ (CC)) = ((AC) ∩ (BC))
42, 3eqtr3i 2375 1 ((AB) ∩ C) = ((AC) ∩ (BC))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642   ∩ cin 3208 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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:  difindir  3510  resindir  4984
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