Theorem eqbrtri 4129

Description: Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
eqbrtr.1	⊢ 𝐴 = 𝐵
eqbrtr.2	⊢ 𝐵𝑅𝐶

Assertion

Ref	Expression
eqbrtri	⊢ 𝐴𝑅𝐶

Proof of Theorem eqbrtri

Step	Hyp	Ref	Expression
1		eqbrtr.2	. 2 ⊢ 𝐵𝑅𝐶
2		eqbrtr.1	. . 3 ⊢ 𝐴 = 𝐵
3	2	breq1i 4115	. 2 ⊢ (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶)
4	1, 3	mpbir 146	1 ⊢ 𝐴𝑅𝐶

Syntax hints:   = wceq 1398   class class class wbr 4108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2814  df-un 3214  df-sn 3694  df-pr 3695  df-op 3697  df-br 4109

This theorem is referenced by:  eqbrtrri  4131  3brtr4i  4138  exmidpw2en  7171  exmidonfinlem  7495  neg1lt0  9344  halflt1  9454  3halfnz  9674  declei  9743  numlti  9744  faclbnd3  11104  geo2lim  12198  0.999...  12203  geoihalfsum  12204  fprodap0  12303  fprodap0f  12318  tan0  12413  cos2bnd  12442  sin4lt0  12449  eirraplem  12459  1nprm  12807  znnen  13141  cnfldstr  14698  tan4thpi  15698  zabsle1  15864  ex-fl  16485  trilpolemisumle  16814