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Theorem repizf2 3906
Description: Replacement. This version of replacement is stronger than repizf 3864 in the sense that does not need to map all values of in to a value of . The resulting set contains those elements for which there is a value of and in that sense, this theorem combines repizf 3864 with ax-sep 3866. Another variation would be  {  |  }  _V but we don't have a proof of that yet. (Contributed by Jim Kingdon, 7-Sep-2018.)
Hypothesis
Ref Expression
repizf2.1  F/
Assertion
Ref Expression
repizf2  {  |  }
Distinct variable group:   ,,,
Allowed substitution hints:   (,,,)

Proof of Theorem repizf2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 vex 2554 . . 3 
_V
21rabex 3892 . 2  {  |  }  _V
3 repizf2lem 3905 . . . 4 
{  |  }
4 nfcv 2175 . . . . . 6  F/_
5 nfrab1 2483 . . . . . 6  F/_ {  |  }
64, 5raleqf 2495 . . . . 5  {  |  } 
{  |  }
7 repizf2.1 . . . . . 6  F/
87repizf 3864 . . . . 5
96, 8syl6bir 153 . . . 4  {  |  }  {  |  }
103, 9syl5bi 141 . . 3  {  |  }
11 df-rab 2309 . . . . . 6  {  |  }  {  |  }
12 nfv 1418 . . . . . . . 8  F/
137nfex 1525 . . . . . . . 8  F/
1412, 13nfan 1454 . . . . . . 7  F/
1514nfab 2179 . . . . . 6  F/_ {  |  }
1611, 15nfcxfr 2172 . . . . 5  F/_ {  |  }
1716nfeq2 2186 . . . 4  F/  {  |  }
184, 5raleqf 2495 . . . 4  {  |  }  {  |  }
1917, 18exbid 1504 . . 3  {  |  } 
{  |  }
2010, 19sylibd 138 . 2  {  |  }  {  |  }
212, 20vtocle 2621 1  {  |  }
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1242   F/wnf 1346  wex 1378  weu 1897  wmo 1898   {cab 2023  wral 2300  wrex 2301   {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  ax-coll 3863  ax-sep 3866
This theorem depends on definitions:  df-bi 110  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-rab 2309  df-v 2553  df-in 2918  df-ss 2925
This theorem is referenced by: (None)
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