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Theorem cbvopab1s 3823
Description: Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 31-Jul-2003.)
Assertion
Ref Expression
cbvopab1s  { <. ,  >.  |  }  { <. ,  >.  |  }
Distinct variable groups:   ,,   ,
Allowed substitution hints:   (,)

Proof of Theorem cbvopab1s
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nfv 1418 . . . 4  F/ 
<. ,  >.
2 nfv 1418 . . . . . 6  F/  <. ,  >.
3 nfs1v 1812 . . . . . 6  F/
42, 3nfan 1454 . . . . 5  F/  <. ,  >.
54nfex 1525 . . . 4  F/ 
<. ,  >.
6 opeq1 3540 . . . . . . 7  <. ,  >.  <. ,  >.
76eqeq2d 2048 . . . . . 6  <. ,  >.  <. ,  >.
8 sbequ12 1651 . . . . . 6
97, 8anbi12d 442 . . . . 5  <. ,  >.  <. ,  >.
109exbidv 1703 . . . 4 
<. ,  >.  <. ,  >.
111, 5, 10cbvex 1636 . . 3  <. , 
>. 
<. ,  >.
1211abbii 2150 . 2  {  |  <. ,  >.  }  {  | 
<. ,  >.  }
13 df-opab 3810 . 2  { <. ,  >.  |  }  {  |  <. ,  >.  }
14 df-opab 3810 . 2  { <. ,  >.  |  }  {  |  <. ,  >.  }
1512, 13, 143eqtr4i 2067 1  { <. ,  >.  |  }  { <. ,  >.  |  }
Colors of variables: wff set class
Syntax hints:   wa 97   wceq 1242  wex 1378  wsb 1642   {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: (None)
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