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Theorem brabsbOLD 4466
Description: The law of concretion in terms of substitutions. (Contributed by NM, 17-Mar-2008.) (New usage is discouraged.) (Proof modification 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 4220 . 2  |-  ( z R w  <->  z { <. x ,  y >.  |  ph } w )
3 df-br 4215 . 2  |-  ( z { <. x ,  y
>.  |  ph } w  <->  <.
z ,  w >.  e. 
{ <. x ,  y
>.  |  ph } )
4 opelopabsbOLD 4465 . 2  |-  ( <.
z ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  [ w  /  y ] [
z  /  x ] ph )
52, 3, 43bitri 264 1  |-  ( z R w  <->  [ w  /  y ] [
z  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1653   [wsb 1659    e. wcel 1726   <.cop 3819   class class class wbr 4214   {copab 4267
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269
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