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Theorem braba 4094
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 4091 . 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 1289    e. wcel 1438   _Vcvv 2619   class class class wbr 3845   {copab 3898
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900
This theorem is referenced by: (None)
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