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Theorem brabg 4629
Description: The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 19-Dec-2013.)
Hypotheses
Ref Expression
opelopabg.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
opelopabg.2  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
brabg.5  |-  R  =  { <. x ,  y
>.  |  ph }
Assertion
Ref Expression
brabg  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A R B  <->  ch ) )
Distinct variable groups:    x, y, A    x, B, y    ch, x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)    C( x, y)    D( x, y)    R( x, y)

Proof of Theorem brabg
StepHypRef Expression
1 opelopabg.1 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
2 opelopabg.2 . . 3  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
31, 2sylan9bb 699 . 2  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ch )
)
4 brabg.5 . 2  |-  R  =  { <. x ,  y
>.  |  ph }
53, 4brabga 4624 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A R B  <->  ch ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   class class class wbr 4313   {copab 4370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pr 4552
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-rab 2745  df-v 2995  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-br 4314  df-opab 4372
This theorem is referenced by:  brab  4632  ideqg  5012  opelcnvg  5040  f1owe  6065  brrpssg  6383  bren  7340  brdomg  7341  brwdom  7803  ltprord  9220  shftfib  12582  efgrelexlema  16267  axcontlem5  23236  cmbr  25009  leopg  25548  cvbr  25708  mdbr  25720  dmdbr  25725  soseq  27737  sltval  27810  isfne  28566  isref  28577  brabg2  28635  isriscg  28816  isfrgra  30608  lcvbr  32762
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