Users' Mathboxes Mathbox for Rodolfo Medina < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  brabsb2 Structured version   Unicode version

Theorem brabsb2 26702
Description: Closed form of brabsbOLD 4456. (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 4206 . . 3  |-  ( R  =  { <. x ,  y >.  |  ph }  ->  ( z R w  <->  z { <. x ,  y >.  |  ph } w ) )
2 df-br 4205 . . 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 4455 . 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 1652   [wsb 1658    e. wcel 1725   <.cop 3809   class class class wbr 4204   {copab 4257
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259
  Copyright terms: Public domain W3C validator