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Theorem indif 3497
 Description: Intersection with class difference. Theorem 34 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
indif (A ∩ (A B)) = (A B)

Proof of Theorem indif
StepHypRef Expression
1 dfin4 3495 . 2 (A ∩ (A B)) = (A (A (A B)))
2 dfin4 3495 . . 3 (AB) = (A (A B))
32difeq2i 3382 . 2 (A (AB)) = (A (A (A B)))
4 difin 3492 . 2 (A (AB)) = (A B)
51, 3, 43eqtr2i 2379 1 (A ∩ (A B)) = (A B)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642   ∖ cdif 3206   ∩ 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-dif 3215  df-ss 3259 This theorem is referenced by: (None)
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