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Theorem dfss4st 3315
 Description: Subclass defined in terms of class difference. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
dfss4st STAB
Distinct variable group:   ,
Allowed substitution hint:   ()

Proof of Theorem dfss4st
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eleq1w 2201 . . . 4
21stbid 818 . . 3 STAB STAB
32cbvalv 1890 . 2 STAB STAB
4 sseqin2 3301 . . 3
5 nfa1 1522 . . . . 5 STAB
6 nfcv 2282 . . . . 5
7 nfcv 2282 . . . . 5
8 eldif 3086 . . . . . . 7
9 eldif 3086 . . . . . . . . . 10
109notbii 658 . . . . . . . . 9
1110anbi2i 453 . . . . . . . 8
12 elin 3265 . . . . . . . . . 10
13 abai 550 . . . . . . . . . 10
1412, 13bitri 183 . . . . . . . . 9
15 imanst 874 . . . . . . . . . 10 STAB
1615anbi2d 460 . . . . . . . . 9 STAB
1714, 16syl5bb 191 . . . . . . . 8 STAB
1811, 17bitr4id 198 . . . . . . 7 STAB
198, 18syl5bb 191 . . . . . 6 STAB
2019sps 1518 . . . . 5 STAB
215, 6, 7, 20eqrd 3121 . . . 4 STAB
2221eqeq1d 2149 . . 3 STAB
234, 22bitr4id 198 . 2 STAB
243, 23sylbi 120 1 STAB
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 103   wb 104  STAB wstab 816  wal 1330   wceq 1332   wcel 1481   cdif 3074   cin 3076   wss 3077 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 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-stab 817  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2692  df-dif 3079  df-in 3083  df-ss 3090 This theorem is referenced by:  sbthlemi3  6857
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