Theorem eqbrtri 4103

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 4089	. 2 ⊢ (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶)
4	1, 3	mpbir 146	1 ⊢ 𝐴𝑅𝐶

Syntax hints:   = wceq 1395   class class class wbr 4082

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 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083

This theorem is referenced by:  eqbrtrri  4105  3brtr4i  4112  exmidpw2en  7062  exmidonfinlem  7359  neg1lt0  9206  halflt1  9316  3halfnz  9532  declei  9601  numlti  9602  faclbnd3  10952  geo2lim  12013  0.999...  12018  geoihalfsum  12019  fprodap0  12118  fprodap0f  12133  tan0  12228  cos2bnd  12257  sin4lt0  12264  eirraplem  12274  1nprm  12622  znnen  12955  cnfldstr  14507  tan4thpi  15500  zabsle1  15663  ex-fl  16019  trilpolemisumle  16337