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Theorem breq12 4209
Description: Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.)
Assertion
Ref Expression
breq12  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A R C  <-> 
B R D ) )

Proof of Theorem breq12
StepHypRef Expression
1 breq1 4207 . 2  |-  ( A  =  B  ->  ( A R C  <->  B R C ) )
2 breq2 4208 . 2  |-  ( C  =  D  ->  ( B R C  <->  B R D ) )
31, 2sylan9bb 681 1  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A R C  <-> 
B R D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652   class class class wbr 4204
This theorem is referenced by:  breq12i  4213  breq12d  4217  breqan12d  4219  posn  4938  isopolem  6057  poxp  6450  soxp  6451  fnse  6455  isprmpt2  6469  ecopover  7000  canth2g  7253  infxpen  7888  sornom  8149  dcomex  8319  zorn2lem6  8373  brdom6disj  8402  fpwwe2  8510  rankcf  8644  ltresr  9007  ltxrlt  9138  wloglei  9551  ltxr  10707  xrltnr  10712  xrltnsym  10722  xrlttri  10724  xrlttr  10725  brfi1uzind  11707  f1olecpbl  13744  isfull  14099  isfth  14103  prslem  14380  pslem  14630  dirtr  14673  xrsdsval  16734  dvcvx  19896  iscusgra  21457  sizeusglecusg  21487  iswlkon  21523  istrlon  21533  ispth  21560  isspth  21561  ispthon  21568  0pthonv  21573  isspthon  21575  1pthon2v  21585  2pthon3v  21596  constr3cyclpe  21642  3v3e3cycl2  21643  iseupa  21679  dfrel4  24026  dfpo2  25370  fununiq  25386  elfix2  25741  axcontlem9  25903  monotoddzzfi  26996  usg2wlk  28272  usg2wlkon  28273
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
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-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205
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