Theorem eqbrtri 4055

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

Hypotheses

Ref	Expression
eqbrtr.1	|- A = B
eqbrtr.2	|- B R C

Assertion

Ref	Expression
eqbrtri	|- A R C

Proof of Theorem eqbrtri

Step	Hyp	Ref	Expression
1		eqbrtr.2	. 2 |- B R C
2		eqbrtr.1	. . 3 |- A = B
3	2	breq1i 4041	. 2 |- ( A R C <-> B R C )
4	1, 3	mpbir 146	1 |- A R C

Syntax hints:   = wceq 1364   class class class wbr 4034

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035

This theorem is referenced by:  eqbrtrri  4057  3brtr4i  4064  exmidpw2en  6982  exmidonfinlem  7272  neg1lt0  9115  halflt1  9225  3halfnz  9440  declei  9509  numlti  9510  faclbnd3  10852  geo2lim  11698  0.999...  11703  geoihalfsum  11704  fprodap0  11803  fprodap0f  11818  tan0  11913  cos2bnd  11942  sin4lt0  11949  eirraplem  11959  1nprm  12307  znnen  12640  cnfldstr  14190  tan4thpi  15161  zabsle1  15324  ex-fl  15455  trilpolemisumle  15769