Theorem eqbrtrri 5142

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

Hypotheses

Ref	Expression
eqbrtrr.1	⊢ 𝐴 = 𝐵
eqbrtrr.2	⊢ 𝐴𝑅𝐶

Assertion

Ref	Expression
eqbrtrri	⊢ 𝐵𝑅𝐶

Proof of Theorem eqbrtrri

Step	Hyp	Ref	Expression
1		eqbrtrr.1	. . 3 ⊢ 𝐴 = 𝐵
2	1	eqcomi 2744	. 2 ⊢ 𝐵 = 𝐴
3		eqbrtrr.2	. 2 ⊢ 𝐴𝑅𝐶
4	2, 3	eqbrtri 5140	1 ⊢ 𝐵𝑅𝐶

Syntax hints:   = wceq 1540   class class class wbr 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120

This theorem is referenced by:  3brtr3i  5148  expnass  14224  faclbnd4lem1  14309  sqrt2gt1lt2  15291  cos1bnd  16203  cos2bnd  16204  prdsvalstr  17464  ovolre  25476  pigt3  26477  pige3ALT  26479  atan1  26888  log2ublem1  26906  sqrtlim  26933  bposlem8  27252  chebbnd1  27433  nohalf  28324  norm-ii-i  31064  nmopadji  32017  unierri  32031  chnub  32938  ballotlem2  34467  hgt750lemd  34626  hgt750lem  34629  stoweidlem26  46003  wallispilem5  46046