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Theorem braba 4050
Description: The law of concretion for a binary relation. (Contributed by NM, 19-Dec-2013.)
Hypotheses
Ref Expression
opelopaba.1  |-  A  e. 
_V
opelopaba.2  |-  B  e. 
_V
opelopaba.3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
braba.4  |-  R  =  { <. x ,  y
>.  |  ph }
Assertion
Ref Expression
braba  |-  ( A R B  <->  ps )
Distinct variable groups:    x, y, A   
x, B, y    ps, x, y
Allowed substitution hints:    ph( x, y)    R( x, y)

Proof of Theorem braba
StepHypRef Expression
1 opelopaba.1 . 2  |-  A  e. 
_V
2 opelopaba.2 . 2  |-  B  e. 
_V
3 opelopaba.3 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
4 braba.4 . . 3  |-  R  =  { <. x ,  y
>.  |  ph }
53, 4brabga 4047 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A R B  <->  ps ) )
61, 2, 5mp2an 417 1  |-  ( A R B  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   _Vcvv 2610   class class class wbr 3805   {copab 3858
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860
This theorem is referenced by: (None)
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