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Theorem difin 3565
 Description: Difference with intersection. Theorem 33 of [Suppes] p. 29. (Contributed by NM, 31-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
difin

Proof of Theorem difin
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 pm4.61 416 . . 3
2 anclb 531 . . . . 5
3 elin 3517 . . . . . 6
43imbi2i 304 . . . . 5
5 iman 414 . . . . 5
62, 4, 53bitr2i 265 . . . 4
76con2bii 323 . . 3
8 eldif 3317 . . 3
91, 7, 83bitr4i 269 . 2
109difeqri 3454 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 359   wceq 1652   wcel 1725   cdif 3304   cin 3306 This theorem is referenced by:  dfin4  3568  indif  3570  symdif1  3593  notrab  3605  dfsdom2  7216  hashdif  11661  isercolllem3  12443  iuncld  17092  llycmpkgen2  17565  1stckgen  17569  ptbasfi  17596  txkgen  17667  cmmbl  19412  disjdifprg2  24001  onint1  26142 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-v 2945  df-dif 3310  df-in 3314
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