Theorem eqbrtrri 5189

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 2749	. 2 ⊢ 𝐵 = 𝐴
3		eqbrtrr.2	. 2 ⊢ 𝐴𝑅𝐶
4	2, 3	eqbrtri 5187	1 ⊢ 𝐵𝑅𝐶

Syntax hints:   = wceq 1537   class class class wbr 5166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167

This theorem is referenced by:  3brtr3i  5195  expnass  14257  faclbnd4lem1  14342  sqrt2gt1lt2  15323  cos1bnd  16235  cos2bnd  16236  2strstr1OLD  17284  prdsvalstr  17512  ovolre  25579  pigt3  26578  pige3ALT  26580  atan1  26989  log2ublem1  27007  sqrtlim  27034  bposlem8  27353  chebbnd1  27534  nohalf  28425  norm-ii-i  31169  nmopadji  32122  unierri  32136  chnub  32984  ballotlem2  34453  hgt750lemd  34625  hgt750lem  34628  stoweidlem26  45947  wallispilem5  45990