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Theorem 3brtr3d 3835
Description: Substitution of equality into both sides of a binary relation. (Contributed by NM, 18-Oct-1999.)
Hypotheses
Ref Expression
3brtr3d.1  |-  ( ph  ->  A R B )
3brtr3d.2  |-  ( ph  ->  A  =  C )
3brtr3d.3  |-  ( ph  ->  B  =  D )
Assertion
Ref Expression
3brtr3d  |-  ( ph  ->  C R D )

Proof of Theorem 3brtr3d
StepHypRef Expression
1 3brtr3d.1 . 2  |-  ( ph  ->  A R B )
2 3brtr3d.2 . . 3  |-  ( ph  ->  A  =  C )
3 3brtr3d.3 . . 3  |-  ( ph  ->  B  =  D )
42, 3breq12d 3819 . 2  |-  ( ph  ->  ( A R B  <-> 
C R D ) )
51, 4mpbid 145 1  |-  ( ph  ->  C R D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285   class class class wbr 3806
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2987  df-sn 3423  df-pr 3424  df-op 3426  df-br 3807
This theorem is referenced by:  ofrval  5774  phplem2  6410  ltaddnq  6695  prarloclemarch2  6707  prmuloclemcalc  6853  axcaucvglemcau  7162  apreap  7790  ltmul1  7795  subap0d  7843  divap1d  7991  lemul2a  8040  monoord2  9588  expubnd  9666  bernneq2  9727  resqrexlemcalc2  10086  resqrexlemcalc3  10087  abs2dif2  10178
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