Theorem eqbrtrri 5134

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 2772	. 2 ⊢ 𝐵 = 𝐴
3		eqbrtrr.2	. 2 ⊢ 𝐴𝑅𝐶
4	2, 3	eqbrtri 5132	1 ⊢ 𝐵𝑅𝐶

This proof depends on syntax axioms:   = wceq 1570   class class class wbr 5108

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109

This theorem is used by:  3brtr3i  5140  expnass  14392  faclbnd4lem1  14478  sqrt2gt1lt2  15484  cos1bnd  16399  cos2bnd  16400  prdsvalstr  17673  chnub  18846  ovolre  25901  pigt3  26903  pige3ALT  26905  atan1  27313  log2ublem1  27331  sqrtlim  27357  bposlem8  27675  chebbnd1  27856  norm-ii-i  31796  nmopadji  32749  unierri  32763  ballotlem2  35232  hgt750lemd  35388  hgt750lem  35391  stoweidlem26  47309  wallispilem5  47352