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Theorem opabbid 3963
Description: Equivalent wff's yield equal ordered-pair class abstractions (deduction form). (Contributed by NM, 21-Feb-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Hypotheses
Ref Expression
opabbid.1  |-  F/ x ph
opabbid.2  |-  F/ y
ph
opabbid.3  |-  ( ph  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
opabbid  |-  ( ph  ->  { <. x ,  y
>.  |  ps }  =  { <. x ,  y
>.  |  ch } )

Proof of Theorem opabbid
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 opabbid.1 . . . 4  |-  F/ x ph
2 opabbid.2 . . . . 5  |-  F/ y
ph
3 opabbid.3 . . . . . 6  |-  ( ph  ->  ( ps  <->  ch )
)
43anbi2d 459 . . . . 5  |-  ( ph  ->  ( ( z  = 
<. x ,  y >.  /\  ps )  <->  ( z  =  <. x ,  y
>.  /\  ch ) ) )
52, 4exbid 1580 . . . 4  |-  ( ph  ->  ( E. y ( z  =  <. x ,  y >.  /\  ps ) 
<->  E. y ( z  =  <. x ,  y
>.  /\  ch ) ) )
61, 5exbid 1580 . . 3  |-  ( ph  ->  ( E. x E. y ( z  = 
<. x ,  y >.  /\  ps )  <->  E. x E. y ( z  = 
<. x ,  y >.  /\  ch ) ) )
76abbidv 2235 . 2  |-  ( ph  ->  { z  |  E. x E. y ( z  =  <. x ,  y
>.  /\  ps ) }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ch ) } )
8 df-opab 3960 . 2  |-  { <. x ,  y >.  |  ps }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ps ) }
9 df-opab 3960 . 2  |-  { <. x ,  y >.  |  ch }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ch ) }
107, 8, 93eqtr4g 2175 1  |-  ( ph  ->  { <. x ,  y
>.  |  ps }  =  { <. x ,  y
>.  |  ch } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1316   F/wnf 1421   E.wex 1453   {cab 2103   <.cop 3500   {copab 3958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-11 1469  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-opab 3960
This theorem is referenced by:  opabbidv  3964  mpteq12f  3978  fnoprabg  5840
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