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Theorem breqtri 4460
Description: Substitution of equal classes into a binary relation. (Contributed by NM, 1-Aug-1999.)
Hypotheses
Ref Expression
breqtr.1  |-  A R B
breqtr.2  |-  B  =  C
Assertion
Ref Expression
breqtri  |-  A R C

Proof of Theorem breqtri
StepHypRef Expression
1 breqtr.1 . 2  |-  A R B
2 breqtr.2 . . 3  |-  B  =  C
32breq2i 4445 . 2  |-  ( A R B  <->  A R C )
41, 3mpbi 208 1  |-  A R C
Colors of variables: wff setvar class
Syntax hints:    = wceq 1383   class class class wbr 4437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438
This theorem is referenced by:  breqtrri  4462  3brtr3i  4464  supsrlem  9491  0lt1  10082  hashunlei  12465  sqrt2gt1lt2  13090  trireciplem  13655  cos1bnd  13904  cos2bnd  13905  cos01gt0  13908  sin4lt0  13912  rpnnen2lem3  13932  gcdaddmlem  14148  dec2dvds  14531  abvtrivd  17468  sincos4thpi  22884  log2cnv  23253  log2ublem2  23256  log2ublem3  23257  birthday  23262  harmonicbnd3  23315  basellem7  23338  ppiublem1  23455  ppiublem2  23456  ppiub  23457  bposlem9  23545  lgsdir2lem2  23577  lgsdir2lem3  23578  ex-fl  25146  siilem1  25744  normlem5  26009  normlem6  26010  norm-ii-i  26032  norm3adifii  26043  cmm2i  26503  mayetes3i  26626  nmopcoadji  26998  mdoc2i  27323  dmdoc2i  27325  sqsscirc1  27868  log2le1  28001  ballotlem1c  28424  lgam1  28584  problem5  29001  circum  29018  cntotbnd  30268  jm2.23  30914  halffl  31447  wallispi  31806  stirlinglem1  31810  fouriersw  31968  bj-pinftyccb  34499  bj-minftyccb  34503
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