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Theorem opabbid 4262
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 685 . . . . 5  |-  ( ph  ->  ( ( z  = 
<. x ,  y >.  /\  ps )  <->  ( z  =  <. x ,  y
>.  /\  ch ) ) )
52, 4exbid 1789 . . . 4  |-  ( ph  ->  ( E. y ( z  =  <. x ,  y >.  /\  ps ) 
<->  E. y ( z  =  <. x ,  y
>.  /\  ch ) ) )
61, 5exbid 1789 . . 3  |-  ( ph  ->  ( E. x E. y ( z  = 
<. x ,  y >.  /\  ps )  <->  E. x E. y ( z  = 
<. x ,  y >.  /\  ch ) ) )
76abbidv 2549 . 2  |-  ( ph  ->  { z  |  E. x E. y ( z  =  <. x ,  y
>.  /\  ps ) }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ch ) } )
8 df-opab 4259 . 2  |-  { <. x ,  y >.  |  ps }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ps ) }
9 df-opab 4259 . 2  |-  { <. x ,  y >.  |  ch }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ch ) }
107, 8, 93eqtr4g 2492 1  |-  ( ph  ->  { <. x ,  y
>.  |  ps }  =  { <. x ,  y
>.  |  ch } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1550   F/wnf 1553    = wceq 1652   {cab 2421   <.cop 3809   {copab 4257
This theorem is referenced by:  opabbidv  4263  mpteq12f  4277  fnoprabg  6163  feqmptdf  24067  mpteq12d  25388
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-opab 4259
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