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Theorem brabg 4387
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 680 . 2  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ch )
)
4 brabg.5 . 2  |-  R  =  { <. x ,  y
>.  |  ph }
53, 4brabga 4382 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A R B  <->  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1647    e. wcel 1715   class class class wbr 4125   {copab 4178
This theorem is referenced by:  brab  4390  ideqg  4938  opelcnvg  4964  f1owe  5973  brrpssg  6421  bren  7014  brdomg  7015  brwdom  7428  ltprord  8801  shftfib  11774  efgrelexlema  15268  cmbr  22597  leopg  23136  cvbr  23296  mdbr  23308  dmdbr  23313  soseq  25080  sltval  25127  axcontlem5  25423  isfne  25860  isref  25871  brabg2  25958  isriscg  26206  isfrgra  27814  lcvbr  29270
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-br 4126  df-opab 4180
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