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Theorem iinab 3709
Description: Indexed intersection of a class builder. (Contributed by NM, 6-Dec-2011.)
Assertion
Ref Expression
iinab  |^|_  {  |  }  {  |  }
Distinct variable groups:   ,   ,
Allowed substitution hints:   (,)   ()

Proof of Theorem iinab
StepHypRef Expression
1 nfcv 2175 . . . 4  F/_
2 nfab1 2177 . . . 4  F/_ {  |  }
31, 2nfiinxy 3675 . . 3  F/_ |^|_  {  |  }
4 nfab1 2177 . . 3  F/_ {  |  }
53, 4cleqf 2198 . 2  |^|_  {  |  }  {  |  } 
|^|_  {  |  }  {  |  }
6 abid 2025 . . . 4  {  |  }
76ralbii 2324 . . 3  {  |  }
8 vex 2554 . . . 4 
_V
9 eliin 3653 . . . 4  _V  |^|_  {  |  }  {  |  }
108, 9ax-mp 7 . . 3  |^|_  {  |  }  {  |  }
11 abid 2025 . . 3  {  |  }
127, 10, 113bitr4i 201 . 2  |^|_  {  |  }  {  |  }
135, 12mpgbir 1339 1  |^|_  {  |  }  {  |  }
Colors of variables: wff set class
Syntax hints:   wb 98   wceq 1242   wcel 1390   {cab 2023  wral 2300   _Vcvv 2551   |^|_ciin 3649
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-bnd 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-ral 2305  df-v 2553  df-iin 3651
This theorem is referenced by:  iinrabm  3710
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