Theorem eqbrtri 4022

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 4008	. 2 |- ( A R C <-> B R C )
4	1, 3	mpbir 146	1 |- A R C

Syntax hints:   = wceq 1353   class class class wbr 4001

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599  df-op 3601  df-br 4002

This theorem is referenced by:  eqbrtrri  4024  3brtr4i  4031  exmidonfinlem  7187  neg1lt0  9021  halflt1  9130  3halfnz  9344  declei  9413  numlti  9414  faclbnd3  10714  geo2lim  11515  0.999...  11520  geoihalfsum  11521  fprodap0  11620  fprodap0f  11635  tan0  11730  cos2bnd  11759  sin4lt0  11765  eirraplem  11775  1nprm  12104  znnen  12389  tan4thpi  14044  zabsle1  14182  ex-fl  14248  trilpolemisumle  14557