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Theorem abrexex2 5693
Description: Existence of an existentially restricted class abstraction. is normally has free-variable parameters and . See also abrexex 5686. (Contributed by NM, 12-Sep-2004.)
Hypotheses
Ref Expression
abrexex2.1  _V
abrexex2.2  {  |  }  _V
Assertion
Ref Expression
abrexex2  {  |  }  _V
Distinct variable group:   ,,
Allowed substitution hints:   (,)

Proof of Theorem abrexex2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nfv 1418 . . . 4  F/
2 nfcv 2175 . . . . 5  F/_
3 nfs1v 1812 . . . . 5  F/
42, 3nfrexxy 2355 . . . 4  F/
5 sbequ12 1651 . . . . 5
65rexbidv 2321 . . . 4
71, 4, 6cbvab 2157 . . 3  {  |  }  {  |  }
8 df-clab 2024 . . . . 5  {  |  }
98rexbii 2325 . . . 4  {  |  }
109abbii 2150 . . 3  {  |  {  |  } }  {  |  }
117, 10eqtr4i 2060 . 2  {  |  }  {  |  {  |  } }
12 df-iun 3650 . . 3  U_  {  |  }  {  |  {  |  } }
13 abrexex2.1 . . . 4  _V
14 abrexex2.2 . . . 4  {  |  }  _V
1513, 14iunex 5692 . . 3  U_  {  |  }  _V
1612, 15eqeltrri 2108 . 2  {  |  {  |  } }  _V
1711, 16eqeltri 2107 1  {  |  }  _V
Colors of variables: wff set class
Syntax hints:   wcel 1390  wsb 1642   {cab 2023  wrex 2301   _Vcvv 2551   U_ciun 3648
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-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853
This theorem is referenced by:  abexssex  5694  abexex  5695  oprabrexex2  5699  ab2rexex  5700  ab2rexex2  5701
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