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Theorem difin 3308
 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 604 . . . . . . . 8
21expd 256 . . . . . . 7
3 dfnot 1349 . . . . . . 7
42, 3syl6ibr 161 . . . . . 6
54com12 30 . . . . 5
65imdistani 441 . . . 4
7 simpr 109 . . . . . 6
87con3i 621 . . . . 5
98anim2i 339 . . . 4
106, 9impbii 125 . . 3
11 eldif 3075 . . . 4
12 elin 3254 . . . . . 6
1312notbii 657 . . . . 5
1413anbi2i 452 . . . 4
1511, 14bitri 183 . . 3
16 eldif 3075 . . 3
1710, 15, 163bitr4i 211 . 2
1817eqriv 2134 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 103   wceq 1331   wfal 1336   wcel 1480   cdif 3063   cin 3065 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-dif 3068  df-in 3072 This theorem is referenced by:  inssddif  3312  symdif1  3336  notrab  3348  disjdif2  3436  unfiin  6807
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