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Theorem difindir 3510
 Description: Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
difindir ((AB) C) = ((A C) ∩ (B C))

Proof of Theorem difindir
StepHypRef Expression
1 inindir 3473 . 2 ((AB) ∩ (V C)) = ((A ∩ (V C)) ∩ (B ∩ (V C)))
2 invdif 3496 . 2 ((AB) ∩ (V C)) = ((AB) C)
3 invdif 3496 . . 3 (A ∩ (V C)) = (A C)
4 invdif 3496 . . 3 (B ∩ (V C)) = (B C)
53, 4ineq12i 3455 . 2 ((A ∩ (V C)) ∩ (B ∩ (V C))) = ((A C) ∩ (B C))
61, 2, 53eqtr3i 2381 1 ((AB) C) = ((A C) ∩ (B C))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642  Vcvv 2859   ∖ cdif 3206   ∩ 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  df-dif 3215 This theorem is referenced by: (None)
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