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Theorem opabbid 3872
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/ 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 452 . . . . 5  |-  ( ph  ->  ( ( z  = 
<. x ,  y >.  /\  ps )  <->  ( z  =  <. x ,  y
>.  /\  ch ) ) )
52, 4exbid 1550 . . . 4  |-  ( ph  ->  ( E. y ( z  =  <. x ,  y >.  /\  ps ) 
<->  E. y ( z  =  <. x ,  y
>.  /\  ch ) ) )
61, 5exbid 1550 . . 3  |-  ( ph  ->  ( E. x E. y ( z  = 
<. x ,  y >.  /\  ps )  <->  E. x E. y ( z  = 
<. x ,  y >.  /\  ch ) ) )
76abbidv 2202 . 2  |-  ( ph  ->  { z  |  E. x E. y ( z  =  <. x ,  y
>.  /\  ps ) }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ch ) } )
8 df-opab 3869 . 2  |-  { <. x ,  y >.  |  ps }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ps ) }
9 df-opab 3869 . 2  |-  { <. x ,  y >.  |  ch }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ch ) }
107, 8, 93eqtr4g 2142 1  |-  ( ph  ->  { <. x ,  y
>.  |  ps }  =  { <. x ,  y
>.  |  ch } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1287   F/wnf 1392   E.wex 1424   {cab 2071   <.cop 3428   {copab 3867
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-11 1440  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-opab 3869
This theorem is referenced by:  opabbidv  3873  mpteq12f  3887  fnoprabg  5684
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