Theorem eqbrtri 4107

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 4093	. 2 ⊢ (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶)
4	1, 3	mpbir 146	1 ⊢ 𝐴𝑅𝐶

Syntax hints:   = wceq 1395   class class class wbr 4086

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2802  df-un 3202  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087

This theorem is referenced by:  eqbrtrri  4109  3brtr4i  4116  exmidpw2en  7099  exmidonfinlem  7397  neg1lt0  9244  halflt1  9354  3halfnz  9570  declei  9639  numlti  9640  faclbnd3  10998  geo2lim  12070  0.999...  12075  geoihalfsum  12076  fprodap0  12175  fprodap0f  12190  tan0  12285  cos2bnd  12314  sin4lt0  12321  eirraplem  12331  1nprm  12679  znnen  13012  cnfldstr  14565  tan4thpi  15558  zabsle1  15721  ex-fl  16271  trilpolemisumle  16592