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Theorem difin2 3365
 Description: Represent a set difference as an intersection with a larger difference. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
difin2

Proof of Theorem difin2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ssel 3118 . . . . 5
21pm4.71d 391 . . . 4
32anbi1d 461 . . 3
4 eldif 3107 . . 3
5 elin 3286 . . . 4
6 eldif 3107 . . . . 5
76anbi1i 454 . . . 4
8 ancom 264 . . . . 5
9 anass 399 . . . . 5
108, 9bitr4i 186 . . . 4
115, 7, 103bitri 205 . . 3
123, 4, 113bitr4g 222 . 2
1312eqrdv 2152 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 103   wceq 1332   wcel 2125   cdif 3095   cin 3097   wss 3098 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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-v 2711  df-dif 3100  df-in 3104  df-ss 3111 This theorem is referenced by: (None)
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