Theorem eqbrtrri 5081

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 2830	. 2 ⊢ 𝐵 = 𝐴
3		eqbrtrr.2	. 2 ⊢ 𝐴𝑅𝐶
4	2, 3	eqbrtri 5079	1 ⊢ 𝐵𝑅𝐶

Syntax hints:   = wceq 1533   class class class wbr 5058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-br 5059

This theorem is referenced by:  3brtr3i  5087  expnass  13564  faclbnd4lem1  13647  sqrt2gt1lt2  14628  cos1bnd  15534  cos2bnd  15535  2strstr1  16599  prdsvalstr  16720  ovolre  24120  pigt3  25097  pige3ALT  25099  atan1  25500  log2ublem1  25518  sqrtlim  25544  bposlem8  25861  chebbnd1  26042  norm-ii-i  28908  nmopadji  29861  unierri  29875  ballotlem2  31741  hgt750lemd  31914  hgt750lem  31917  stoweidlem26  42305  wallispilem5  42348