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Theorem brabsb2 26833
Description: Closed form of brabsbOLD 4290. (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 4041 . . 3  |-  ( R  =  { <. x ,  y >.  |  ph }  ->  ( z R w  <->  z { <. x ,  y >.  |  ph } w ) )
2 df-br 4040 . . 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 4289 . 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 1632   [wsb 1638    e. wcel 1696   <.cop 3656   class class class wbr 4039   {copab 4092
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094
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