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Theorem difindi 3509
 Description: Distributive law for class difference. Theorem 40 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
difindi (A (BC)) = ((A B) ∪ (A C))

Proof of Theorem difindi
StepHypRef Expression
1 dfin3 3494 . . 3 (BC) = (V ((V B) ∪ (V C)))
21difeq2i 3382 . 2 (A (BC)) = (A (V ((V B) ∪ (V C))))
3 indi 3501 . . 3 (A ∩ ((V B) ∪ (V C))) = ((A ∩ (V B)) ∪ (A ∩ (V C)))
4 dfin2 3491 . . 3 (A ∩ ((V B) ∪ (V C))) = (A (V ((V B) ∪ (V C))))
5 invdif 3496 . . . 4 (A ∩ (V B)) = (A B)
6 invdif 3496 . . . 4 (A ∩ (V C)) = (A C)
75, 6uneq12i 3416 . . 3 ((A ∩ (V B)) ∪ (A ∩ (V C))) = ((A B) ∪ (A C))
83, 4, 73eqtr3i 2381 . 2 (A (V ((V B) ∪ (V C)))) = ((A B) ∪ (A C))
92, 8eqtri 2373 1 (A (BC)) = ((A B) ∪ (A C))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642  Vcvv 2859   ∖ cdif 3206   ∪ cun 3207   ∩ 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  df-un 3214  df-dif 3215 This theorem is referenced by:  indm  3513
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