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Theorem brabsb2 26142
Description: Closed form of brabsbOLD 4274. (Contributed by Rodolfo Medina, 13-Oct-2010.)
Assertion
Ref Expression
brabsb2  |-  ( R  =  { <. x ,  y >.  |  ph }  ->  ( z R w  <->  [ w  /  y ] [ z  /  x ] ph ) )
Distinct variable groups:    x, y,
z    x, w, y
Allowed substitution hints:    ph( x, y, z, w)    R( x, y, z, w)

Proof of Theorem brabsb2
StepHypRef Expression
1 breq 4025 . . 3  |-  ( R  =  { <. x ,  y >.  |  ph }  ->  ( z R w  <->  z { <. x ,  y >.  |  ph } w ) )
2 df-br 4024 . . 3  |-  ( z { <. x ,  y
>.  |  ph } w  <->  <.
z ,  w >.  e. 
{ <. x ,  y
>.  |  ph } )
31, 2syl6bb 252 . 2  |-  ( R  =  { <. x ,  y >.  |  ph }  ->  ( z R w  <->  <. z ,  w >.  e.  { <. x ,  y >.  |  ph } ) )
4 opelopabsbOLD 4273 . 2  |-  ( <.
z ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  [ w  /  y ] [
z  /  x ] ph )
53, 4syl6bb 252 1  |-  ( R  =  { <. x ,  y >.  |  ph }  ->  ( z R w  <->  [ w  /  y ] [ z  /  x ] ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623   [wsb 1629    e. wcel 1684   <.cop 3643   class class class wbr 4023   {copab 4076
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078
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