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Theorem braba 4472
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 4469 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A R B  <->  ps ) )
61, 2, 5mp2an 654 1  |-  ( A R B  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956   class class class wbr 4212   {copab 4265
This theorem is referenced by:  frgpuplem  15404  2ndcctbss  17518  prtlem13  26717  wepwsolem  27116  fnwe2val  27124
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 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
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-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267
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