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Theorem breqan12d 4227
Description: Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
Hypotheses
Ref Expression
breq1d.1  |-  ( ph  ->  A  =  B )
breqan12i.2  |-  ( ps 
->  C  =  D
)
Assertion
Ref Expression
breqan12d  |-  ( (
ph  /\  ps )  ->  ( A R C  <-> 
B R D ) )

Proof of Theorem breqan12d
StepHypRef Expression
1 breq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 breqan12i.2 . 2  |-  ( ps 
->  C  =  D
)
3 breq12 4217 . 2  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A R C  <-> 
B R D ) )
41, 2, 3syl2an 464 1  |-  ( (
ph  /\  ps )  ->  ( 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 4212
This theorem is referenced by:  breqan12rd  4228  soisores  6047  isoid  6049  isores3  6055  isoini2  6059  ofrfval  6313  fnwelem  6461  fnse  6463  ovec  7014  wemaplem1  7515  r0weon  7894  sornom  8157  enqbreq2  8797  nqereu  8806  ordpinq  8820  lterpq  8847  ltresr2  9016  axpre-ltadd  9042  leltadd  9512  lemul1a  9864  negiso  9984  xltneg  10803  lt2sq  11455  le2sq  11456  sqrle  12066  prdsleval  13699  efgcpbllema  15386  iducn  18313  icopnfhmeo  18968  iccpnfhmeo  18970  xrhmeo  18971  reefiso  20364  sinord  20436  logltb  20494  logccv  20554  atanord  20767  birthdaylem3  20792  lgsquadlem3  21140  mddmd  23804  xrge0iifiso  24321  erdszelem4  24880  erdszelem8  24884  cgrextend  25942  monotuz  27004  monotoddzzfi  27005  expmordi  27010  wepwsolem  27116  fnwe2val  27124  aomclem8  27136  idlaut  30893
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 2417
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 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  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
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