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Theorem breqtr 2628
Description: Substitution of equal classes into a binary relation.
Hypotheses
Ref Expression
breqtr.1 |- ARB
breqtr.2 |- B = C
Assertion
Ref Expression
breqtr |- ARC
Proof of Theorem breqtr
StepHypRef Expression
1 breqtr.1 . 2 |- ARB
2 breqtr.2 . . 3 |- B = C
32breq2i 2617 . 2 |- (ARB <-> ARC)
41, 3mpbi 189 1 |- ARC
Colors of variables: wff set class
Syntax hints:   = wceq 953   class class class wbr 2609
This theorem is referenced by:  breqtrr 2630  3brtr3 2632  cdacomen 4901  cdaassen 4902  lt01 5653  sqrlem10 6612  sqrlem11 6613  sqr2gt1lt2 6649  abslt 6810  absle 6811  absltOLD 6812  absleOLD 6813  abstri 6829  infcvglem2 7157  expcnvlem2 7163  geolimilem 7170  erelem2 7262  efaddlem16 7295  ef1tllem 7323  eirrlem3 7332  cos1bnd 7416  cos2bnd 7417  cos01gt0 7419  sin4lt0 7423  ruclem30 7482  siilem1 8442  sincos4thpi 8627  cosh111lem1 8629  normlem5 8901  normlem6 8902  norm-ii 8925  norm3adif 8936  projlem3 9104  projlem18 9119  cmm2 9467  nmopcoadj 9948  mdoc2 10258  dmdoc2 10260
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-un 2040  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610
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