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Theorem brabsbOLD 4311
Description: The law of concretion in terms of substitutions. (Contributed by NM, 17-Mar-2008.) (New usage is discouraged.)
Hypothesis
Ref Expression
brabsbOLD.1  |-  R  =  { <. x ,  y
>.  |  ph }
Assertion
Ref Expression
brabsbOLD  |-  ( 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 brabsbOLD
StepHypRef Expression
1 brabsbOLD.1 . . 3  |-  R  =  { <. x ,  y
>.  |  ph }
21breqi 4066 . 2  |-  ( z R w  <->  z { <. x ,  y >.  |  ph } w )
3 df-br 4061 . 2  |-  ( z { <. x ,  y
>.  |  ph } w  <->  <.
z ,  w >.  e. 
{ <. x ,  y
>.  |  ph } )
4 opelopabsbOLD 4310 . 2  |-  ( <.
z ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  [ w  /  y ] [
z  /  x ] ph )
52, 3, 43bitri 262 1  |-  ( z R w  <->  [ w  /  y ] [
z  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1633   [wsb 1639    e. wcel 1701   <.cop 3677   class class class wbr 4060   {copab 4113
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-br 4061  df-opab 4115
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