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Theorem elrab3t 2691
Description: Membership in a restricted class abstraction, using implicit substitution. (Closed theorem version of elrab3 2693.) (Contributed by Thierry Arnoux, 31-Aug-2017.)
Assertion
Ref Expression
elrab3t  {  |  }
Distinct variable groups:   ,   ,   ,
Allowed substitution hint:   ()

Proof of Theorem elrab3t
StepHypRef Expression
1 simpr 103 . . 3
2 nfa1 1431 . . . . 5  F/
3 nfv 1418 . . . . 5  F/
42, 3nfan 1454 . . . 4  F/
5 simpl 102 . . . . . 6
6519.21bi 1447 . . . . 5
7 eleq1 2097 . . . . . . . . . 10
87biimparc 283 . . . . . . . . 9
98biantrurd 289 . . . . . . . 8
109bibi1d 222 . . . . . . 7
1110pm5.74da 417 . . . . . 6
1211adantl 262 . . . . 5
136, 12mpbid 135 . . . 4
144, 13alrimi 1412 . . 3
15 elabgt 2678 . . 3  {  |  }
161, 14, 15syl2anc 391 . 2  {  |  }
17 df-rab 2309 . . . 4  {  |  }  {  |  }
1817eleq2i 2101 . . 3  {  |  }  {  |  }
1918bibi1i 217 . 2  {  |  }  {  |  }
2016, 19sylibr 137 1  {  |  }
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98  wal 1240   wceq 1242   wcel 1390   {cab 2023   {crab 2304
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rab 2309  df-v 2553
This theorem is referenced by:  f1oresrab  5272
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