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Theorem brabsb2 26395
Description: Closed form of brabsbOLD 4398. (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 4148 . . 3  |-  ( R  =  { <. x ,  y >.  |  ph }  ->  ( z R w  <->  z { <. x ,  y >.  |  ph } w ) )
2 df-br 4147 . . 3  |-  ( z { <. x ,  y
>.  |  ph } w  <->  <.
z ,  w >.  e. 
{ <. x ,  y
>.  |  ph } )
31, 2syl6bb 253 . 2  |-  ( R  =  { <. x ,  y >.  |  ph }  ->  ( z R w  <->  <. z ,  w >.  e.  { <. x ,  y >.  |  ph } ) )
4 opelopabsbOLD 4397 . 2  |-  ( <.
z ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  [ w  /  y ] [
z  /  x ] ph )
53, 4syl6bb 253 1  |-  ( R  =  { <. x ,  y >.  |  ph }  ->  ( z R w  <->  [ w  /  y ] [ z  /  x ] ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649   [wsb 1655    e. wcel 1717   <.cop 3753   class class class wbr 4146   {copab 4199
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-br 4147  df-opab 4201
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