Theorem eqbrtrri 4991

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 2806	. 2 ⊢ 𝐵 = 𝐴
3		eqbrtrr.2	. 2 ⊢ 𝐴𝑅𝐶
4	2, 3	eqbrtri 4989	1 ⊢ 𝐵𝑅𝐶

Syntax hints:   = wceq 1525   class class class wbr 4968

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-rab 3116  df-v 3442  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-br 4969

This theorem is referenced by:  3brtr3i  4997  expnass  13424  faclbnd4lem1  13507  sqrt2gt1lt2  14472  cos1bnd  15377  cos2bnd  15378  2strstr1  16438  prdsvalstr  16559  ovolre  23813  pigt3  24790  pige3ALT  24792  atan1  25191  log2ublem1  25210  sqrtlim  25236  bposlem8  25553  chebbnd1  25734  norm-ii-i  28601  nmopadji  29554  unierri  29568  ballotlem2  31359  hgt750lemd  31532  hgt750lem  31535  stoweidlem26  41875  wallispilem5  41918