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Theorem difun1 3514
 Description: A relationship involving double difference and union. (Contributed by NM, 29-Aug-2004.)
Assertion
Ref Expression
difun1 (A (BC)) = ((A B) C)

Proof of Theorem difun1
StepHypRef Expression
1 inass 3465 . . . 4 ((A ∩ (V B)) ∩ (V C)) = (A ∩ ((V B) ∩ (V C)))
2 invdif 3496 . . . 4 ((A ∩ (V B)) ∩ (V C)) = ((A ∩ (V B)) C)
31, 2eqtr3i 2375 . . 3 (A ∩ ((V B) ∩ (V C))) = ((A ∩ (V B)) C)
4 undm 3512 . . . . 5 (V (BC)) = ((V B) ∩ (V C))
54ineq2i 3454 . . . 4 (A ∩ (V (BC))) = (A ∩ ((V B) ∩ (V C)))
6 invdif 3496 . . . 4 (A ∩ (V (BC))) = (A (BC))
75, 6eqtr3i 2375 . . 3 (A ∩ ((V B) ∩ (V C))) = (A (BC))
83, 7eqtr3i 2375 . 2 ((A ∩ (V B)) C) = (A (BC))
9 invdif 3496 . . 3 (A ∩ (V B)) = (A B)
109difeq1i 3381 . 2 ((A ∩ (V B)) C) = ((A B) C)
118, 10eqtr3i 2375 1 (A (BC)) = ((A B) 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:  dif32  3517  difabs  3518
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