Theorem breqtri 4678

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

Step	Hyp	Ref	Expression
1		breqtr.1	. 2 |- A R B
2		breqtr.2	. . 3 |- B = C
3	2	breq2i 4661	. 2 |- ( A R B <-> A R C )
4	1, 3	mpbi 220	1 |- A R C

Syntax hints:   = wceq 1483   class class class wbr 4653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654

This theorem is referenced by:  breqtrri  4680  3brtr3i  4682  supsrlem  9932  0lt1  10550  le9lt10  11529  9lt10  11673  hashunlei  13212  sqrt2gt1lt2  14015  trireciplem  14594  cos1bnd  14917  cos2bnd  14918  cos01gt0  14921  sin4lt0  14925  rpnnen2lem3  14945  z4even  15108  gcdaddmlem  15245  dec2dvds  15767  abvtrivd  18840  sincos4thpi  24265  log2cnv  24671  log2ublem2  24674  log2ublem3  24675  log2le1  24677  birthday  24681  harmonicbnd3  24734  lgam1  24790  basellem7  24813  ppiublem1  24927  ppiublem2  24928  ppiub  24929  bposlem4  25012  bposlem5  25013  bposlem9  25017  lgsdir2lem2  25051  lgsdir2lem3  25052  ex-fl  27304  siilem1  27706  normlem5  27971  normlem6  27972  norm-ii-i  27994  norm3adifii  28005  cmm2i  28466  mayetes3i  28588  nmopcoadji  28960  mdoc2i  29285  dmdoc2i  29287  dp2lt10  29591  dp2ltsuc  29593  dplti  29613  sqsscirc1  29954  ballotlem1c  30569  hgt750lem  30729  problem5  31563  circum  31568  bj-pinftyccb  33108  bj-minftyccb  33112  poimirlem25  33434  cntotbnd  33595  jm2.23  37563  halffl  39510  wallispi  40287  stirlinglem1  40291  fouriersw  40448