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Theorem cbvopab 3819
Description: Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 14-Sep-2003.)
Hypotheses
Ref Expression
cbvopab.1  F/
cbvopab.2  F/
cbvopab.3  F/
cbvopab.4  F/
cbvopab.5
Assertion
Ref Expression
cbvopab  { <. ,  >.  |  }  { <. ,  >.  |  }
Distinct variable group:   ,,,
Allowed substitution hints:   (,,,)   (,,,)

Proof of Theorem cbvopab
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nfv 1418 . . . . 5  F/  <. ,  >.
2 cbvopab.1 . . . . 5  F/
31, 2nfan 1454 . . . 4  F/  <. ,  >.
4 nfv 1418 . . . . 5  F/  <. ,  >.
5 cbvopab.2 . . . . 5  F/
64, 5nfan 1454 . . . 4  F/  <. ,  >.
7 nfv 1418 . . . . 5  F/  <. ,  >.
8 cbvopab.3 . . . . 5  F/
97, 8nfan 1454 . . . 4  F/  <. ,  >.
10 nfv 1418 . . . . 5  F/  <. ,  >.
11 cbvopab.4 . . . . 5  F/
1210, 11nfan 1454 . . . 4  F/  <. ,  >.
13 opeq12 3542 . . . . . 6  <. , 
>.  <. ,  >.
1413eqeq2d 2048 . . . . 5  <. ,  >. 
<. ,  >.
15 cbvopab.5 . . . . 5
1614, 15anbi12d 442 . . . 4 
<. ,  >.  <. ,  >.
173, 6, 9, 12, 16cbvex2 1794 . . 3  <. , 
>. 
<. ,  >.
1817abbii 2150 . 2  {  |  <. ,  >.  }  {  | 
<. ,  >.  }
19 df-opab 3810 . 2  { <. ,  >.  |  }  {  |  <. ,  >.  }
20 df-opab 3810 . 2  { <. ,  >.  |  }  {  |  <. ,  >.  }
2118, 19, 203eqtr4i 2067 1  { <. ,  >.  |  }  { <. ,  >.  |  }
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242   F/wnf 1346  wex 1378   {cab 2023   <.cop 3370   {copab 3808
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-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-opab 3810
This theorem is referenced by:  cbvopabv  3820  opelopabsb  3988
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