Theorem eqbrtrri 5166

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 2746	. 2 ⊢ 𝐵 = 𝐴
3		eqbrtrr.2	. 2 ⊢ 𝐴𝑅𝐶
4	2, 3	eqbrtri 5164	1 ⊢ 𝐵𝑅𝐶

Syntax hints:   = wceq 1540   class class class wbr 5143

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 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144

This theorem is referenced by:  3brtr3i  5172  expnass  14247  faclbnd4lem1  14332  sqrt2gt1lt2  15313  cos1bnd  16223  cos2bnd  16224  2strstr1OLD  17271  prdsvalstr  17497  ovolre  25560  pigt3  26560  pige3ALT  26562  atan1  26971  log2ublem1  26989  sqrtlim  27016  bposlem8  27335  chebbnd1  27516  nohalf  28407  norm-ii-i  31156  nmopadji  32109  unierri  32123  chnub  33002  ballotlem2  34491  hgt750lemd  34663  hgt750lem  34666  stoweidlem26  46041  wallispilem5  46084