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Theorem eqbrtrrd 3809
Description: Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
Hypotheses
Ref Expression
eqbrtrrd.1  |-  ( ph  ->  A  =  B )
eqbrtrrd.2  |-  ( ph  ->  A R C )
Assertion
Ref Expression
eqbrtrrd  |-  ( ph  ->  B R C )

Proof of Theorem eqbrtrrd
StepHypRef Expression
1 eqbrtrrd.1 . . 3  |-  ( ph  ->  A  =  B )
21eqcomd 2087 . 2  |-  ( ph  ->  B  =  A )
3 eqbrtrrd.2 . 2  |-  ( ph  ->  A R C )
42, 3eqbrtrd 3807 1  |-  ( ph  ->  B R C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285   class class class wbr 3787
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 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-sn 3406  df-pr 3407  df-op 3409  df-br 3788
This theorem is referenced by:  dftpos4  5906  phpm  6390  unsnfidcex  6430  prmuloclemcalc  6806  mullocprlem  6811  cauappcvgprlemladdfl  6896  caucvgprlemopl  6910  caucvgprprlemloccalc  6925  caucvgprprlemopl  6938  ltadd1sr  7004  axarch  7108  lemulge11  8000  modqmuladdim  9438  ltexp2a  9614  leexp2a  9615  nnlesq  9664  faclbnd6  9757  facavg  9759  cvg1nlemcxze  9995  resqrexlemover  10023  resqrexlemlo  10026  resqrexlemnmsq  10030  resqrexlemnm  10031  leabs  10087  abs3dif  10118  abs2dif  10119  maxabslemlub  10220  maxltsup  10231  recn2  10282  imcn2  10283  iiserex  10304  divalglemnqt  10453  mulgcd  10538  dvdssqlem  10552  nn0seqcvgd  10556  mulgcddvds  10609  rpdvds  10614  pw2dvdseulemle  10678  sqrt2irraplemnn  10690
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