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Theorem 3brtr4d 3875
Description: Substitution of equality into both sides of a binary relation. (Contributed by NM, 21-Feb-2005.)
Hypotheses
Ref Expression
3brtr4d.1  |-  ( ph  ->  A R B )
3brtr4d.2  |-  ( ph  ->  C  =  A )
3brtr4d.3  |-  ( ph  ->  D  =  B )
Assertion
Ref Expression
3brtr4d  |-  ( ph  ->  C R D )

Proof of Theorem 3brtr4d
StepHypRef Expression
1 3brtr4d.1 . 2  |-  ( ph  ->  A R B )
2 3brtr4d.2 . . 3  |-  ( ph  ->  C  =  A )
3 3brtr4d.3 . . 3  |-  ( ph  ->  D  =  B )
42, 3breq12d 3858 . 2  |-  ( ph  ->  ( C R D  <-> 
A R B ) )
51, 4mpbird 165 1  |-  ( ph  ->  C R D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289   class class class wbr 3845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846
This theorem is referenced by:  f1oiso2  5606  prarloclemarch2  6976  caucvgprprlemmu  7252  caucvgsrlembound  7337  mulap0  8121  lediv12a  8353  recp1lt1  8358  fldiv4p1lem1div2  9708  intfracq  9723  modqmulnn  9745  addmodlteq  9801  frecfzennn  9829  monoord2  9901  expgt1  9989  leexp2r  10005  leexp1a  10006  bernneq  10070  faclbnd  10145  faclbnd6  10148  facubnd  10149  hashunlem  10208  zfz1isolemiso  10240  sqrtgt0  10463  absrele  10512  absimle  10513  abstri  10533  abs2difabs  10537  climsqz  10719  climsqz2  10720  fisumcvg2  10782  fsum3cvg2  10783  isumle  10885  expcnvap0  10892  expcnvre  10893  explecnv  10895  cvgratz  10922  efcllemp  10944  ege2le3  10957  eflegeo  10988  phibnd  11467
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