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Theorem difin 3217
 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 ax-in2 578 . . . . . . . 8
21expd 254 . . . . . . 7
3 dfnot 1303 . . . . . . 7
42, 3syl6ibr 160 . . . . . 6
54com12 30 . . . . 5
65imdistani 434 . . . 4
7 simpr 108 . . . . . 6
87con3i 595 . . . . 5
98anim2i 334 . . . 4
106, 9impbii 124 . . 3
11 eldif 2991 . . . 4
12 elin 3165 . . . . . 6
1312notbii 627 . . . . 5
1413anbi2i 445 . . . 4
1511, 14bitri 182 . . 3
16 eldif 2991 . . 3
1710, 15, 163bitr4i 210 . 2
1817eqriv 2080 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 102   wceq 1285   wfal 1290   wcel 1434   cdif 2979   cin 2981 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-dif 2984  df-in 2988 This theorem is referenced by:  inssddif  3221  symdif1  3245  notrab  3257  unfiin  6470
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