Theorem breqtri 5055

Description: Substitution of equal classes into a binary relation. (Contributed by NM, 1-Aug-1999.)

Hypotheses

Ref	Expression
breqtr.1	⊢ 𝐴𝑅𝐵
breqtr.2	⊢ 𝐵 = 𝐶

Assertion

Ref	Expression
breqtri	⊢ 𝐴𝑅𝐶

Proof of Theorem breqtri

Step	Hyp	Ref	Expression
1		breqtr.1	. 2 ⊢ 𝐴𝑅𝐵
2		breqtr.2	. . 3 ⊢ 𝐵 = 𝐶
3	2	breq2i 5038	. 2 ⊢ (𝐴𝑅𝐵 ↔ 𝐴𝑅𝐶)
4	1, 3	mpbi 233	1 ⊢ 𝐴𝑅𝐶

Syntax hints:   = wceq 1538   class class class wbr 5030

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-un 3886  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031

This theorem is referenced by:  breqtrri  5057  3brtr3i  5059  supsrlem  10522  0lt1  11151  le9lt10  12113  9lt10  12217  hashunlei  13782  sqrt2gt1lt2  14626  trireciplem  15209  cos1bnd  15532  cos2bnd  15533  cos01gt0  15536  sin4lt0  15540  rpnnen2lem3  15561  z4even  15713  gcdaddmlem  15862  dec2dvds  16389  abvtrivd  19604  sincos4thpi  25106  log2cnv  25530  log2ublem2  25533  log2ublem3  25534  log2le1  25536  birthday  25540  harmonicbnd3  25593  lgam1  25649  basellem7  25672  ppiublem1  25786  ppiub  25788  bposlem4  25871  bposlem5  25872  bposlem9  25876  lgsdir2lem2  25910  lgsdir2lem3  25911  ex-fl  28232  siilem1  28634  normlem5  28897  normlem6  28898  norm-ii-i  28920  norm3adifii  28931  cmm2i  29390  mayetes3i  29512  nmopcoadji  29884  mdoc2i  30209  dmdoc2i  30211  dp2lt10  30586  dp2ltsuc  30588  dplti  30607  sqsscirc1  31261  ballotlem1c  31875  hgt750lem  32032  problem5  33025  circum  33030  bj-pinftyccb  34636  bj-minftyccb  34640  poimirlem25  35082  cntotbnd  35234  3lexlogpow5ineq1  39341  jm2.23  39937  tr3dom  40236  halffl  41928  wallispi  42712  stirlinglem1  42716  fouriersw  42873