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Theorem breqtrrd 3790
Description: Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
Hypotheses
Ref Expression
breqtrrd.1  |-  ( ph  ->  A R B )
breqtrrd.2  |-  ( ph  ->  C  =  B )
Assertion
Ref Expression
breqtrrd  |-  ( ph  ->  A R C )

Proof of Theorem breqtrrd
StepHypRef Expression
1 breqtrrd.1 . 2  |-  ( ph  ->  A R B )
2 breqtrrd.2 . . 3  |-  ( ph  ->  C  =  B )
32eqcomd 2045 . 2  |-  ( ph  ->  B  =  C )
41, 3breqtrd 3788 1  |-  ( ph  ->  A R C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1243   class class class wbr 3764
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765
This theorem is referenced by:  frirrg  4087  addlocprlemeq  6631  ltexprlemopl  6699  recexprlemloc  6729  cauappcvgprlemopl  6744  cauappcvgprlemladdfu  6752  cauappcvgprlem1  6757  caucvgprlemopl  6767  caucvgprlemladdfu  6775  caucvgprprlemopl  6795  caucvgprprlemexbt  6804  mulgt0sr  6862  archsr  6866  caucvgsrlemoffgt1  6883  mulap0r  7606  prodgt0  7818  div4p1lem1div2  8177  uzsubsubfz  8911  fzctr  8991  subfzo0  9097  qbtwnzlemstep  9103  qbtwnzlemex  9105  rebtwn2zlemstep  9107  rebtwn2z  9109  ceilqge  9152  modqge0  9174  modqlt  9175  isermono  9237  serile  9253  leexp1a  9309  sqgt0ap  9322  sqge0  9330  nnlesq  9356  expnbnd  9372  cjmulge0  9489  resqrexlemover  9608  resqrexlemdec  9609  resqrexlemlo  9611  resqrexlemcalc3  9614  resqrexlemcvg  9617  resqrexlemoverl  9619  resqrexlemglsq  9620  resqrexlemga  9621  absge0  9658  amgm2  9714  climle  9854  climserile  9865  nn0seqcvgd  9880
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