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Theorem opabbid 4449
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 703 . . . . 5  |-  ( ph  ->  ( ( z  = 
<. x ,  y >.  /\  ps )  <->  ( z  =  <. x ,  y
>.  /\  ch ) ) )
52, 4exbid 1822 . . . 4  |-  ( ph  ->  ( E. y ( z  =  <. x ,  y >.  /\  ps ) 
<->  E. y ( z  =  <. x ,  y
>.  /\  ch ) ) )
61, 5exbid 1822 . . 3  |-  ( ph  ->  ( E. x E. y ( z  = 
<. x ,  y >.  /\  ps )  <->  E. x E. y ( z  = 
<. x ,  y >.  /\  ch ) ) )
76abbidv 2585 . 2  |-  ( ph  ->  { z  |  E. x E. y ( z  =  <. x ,  y
>.  /\  ps ) }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ch ) } )
8 df-opab 4446 . 2  |-  { <. x ,  y >.  |  ps }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ps ) }
9 df-opab 4446 . 2  |-  { <. x ,  y >.  |  ch }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ch ) }
107, 8, 93eqtr4g 2516 1  |-  ( ph  ->  { <. x ,  y
>.  |  ps }  =  { <. x ,  y
>.  |  ch } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370   E.wex 1587   F/wnf 1590   {cab 2436   <.cop 3978   {copab 4444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-opab 4446
This theorem is referenced by:  opabbidv  4450  mpteq12f  4463  fnoprabg  6288  feqmptdf  26109  mpteq12d  27714
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