Theorem eqbrtrri 5170

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 2743	. 2 ⊢ 𝐵 = 𝐴
3		eqbrtrr.2	. 2 ⊢ 𝐴𝑅𝐶
4	2, 3	eqbrtri 5168	1 ⊢ 𝐵𝑅𝐶

Syntax hints:   = wceq 1536   class class class wbr 5147

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148

This theorem is referenced by:  3brtr3i  5176  expnass  14243  faclbnd4lem1  14328  sqrt2gt1lt2  15309  cos1bnd  16219  cos2bnd  16220  2strstr1OLD  17270  prdsvalstr  17498  ovolre  25573  pigt3  26574  pige3ALT  26576  atan1  26985  log2ublem1  27003  sqrtlim  27030  bposlem8  27349  chebbnd1  27530  nohalf  28421  norm-ii-i  31165  nmopadji  32118  unierri  32132  chnub  32985  ballotlem2  34469  hgt750lemd  34641  hgt750lem  34644  stoweidlem26  45981  wallispilem5  46024