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Theorem opabbid 3813
Description: Equivalent wff's yield equal ordered-pair class abstractions (deduction rule). (Contributed by NM, 21-Feb-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Hypotheses
Ref Expression
opabbid.1  F/
opabbid.2  F/
opabbid.3
Assertion
Ref Expression
opabbid  { <. , 
>.  |  }  { <. , 
>.  |  }

Proof of Theorem opabbid
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 opabbid.1 . . . 4  F/
2 opabbid.2 . . . . 5  F/
3 opabbid.3 . . . . . 6
43anbi2d 437 . . . . 5 
<. ,  >.  <. , 
>.
52, 4exbid 1504 . . . 4  <. ,  >. 
<. ,  >.
61, 5exbid 1504 . . 3  <. ,  >.  <. ,  >.
76abbidv 2152 . 2  {  |  <. , 
>.  }  {  |  <. , 
>.  }
8 df-opab 3810 . 2  { <. ,  >.  |  }  {  |  <. ,  >.  }
9 df-opab 3810 . 2  { <. ,  >.  |  }  {  |  <. ,  >.  }
107, 8, 93eqtr4g 2094 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-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  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-opab 3810
This theorem is referenced by:  opabbidv  3814  mpteq12f  3828  fnoprabg  5544
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