Theorem breqtri 5091

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 5074	. 2 ⊢ (𝐴𝑅𝐵 ↔ 𝐴𝑅𝐶)
4	1, 3	mpbi 232	1 ⊢ 𝐴𝑅𝐶

Syntax hints:   = wceq 1537   class class class wbr 5066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067

This theorem is referenced by:  breqtrri  5093  3brtr3i  5095  supsrlem  10533  0lt1  11162  le9lt10  12126  9lt10  12230  hashunlei  13787  sqrt2gt1lt2  14634  trireciplem  15217  cos1bnd  15540  cos2bnd  15541  cos01gt0  15544  sin4lt0  15548  rpnnen2lem3  15569  z4even  15723  gcdaddmlem  15872  dec2dvds  16399  abvtrivd  19611  sincos4thpi  25099  log2cnv  25522  log2ublem2  25525  log2ublem3  25526  log2le1  25528  birthday  25532  harmonicbnd3  25585  lgam1  25641  basellem7  25664  ppiublem1  25778  ppiub  25780  bposlem4  25863  bposlem5  25864  bposlem9  25868  lgsdir2lem2  25902  lgsdir2lem3  25903  ex-fl  28226  siilem1  28628  normlem5  28891  normlem6  28892  norm-ii-i  28914  norm3adifii  28925  cmm2i  29384  mayetes3i  29506  nmopcoadji  29878  mdoc2i  30203  dmdoc2i  30205  dp2lt10  30560  dp2ltsuc  30562  dplti  30581  sqsscirc1  31151  ballotlem1c  31765  hgt750lem  31922  problem5  32912  circum  32917  bj-pinftyccb  34506  bj-minftyccb  34510  poimirlem25  34932  cntotbnd  35089  jm2.23  39613  tr3dom  39914  halffl  41583  wallispi  42375  stirlinglem1  42379  fouriersw  42536