Theorem eqbrtri 4024

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 4010	. 2 |- ( A R C <-> B R C )
4	1, 3	mpbir 146	1 |- A R C

Syntax hints:   = wceq 1353   class class class wbr 4003

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 4004

This theorem is referenced by:  eqbrtrri  4026  3brtr4i  4033  exmidonfinlem  7191  neg1lt0  9026  halflt1  9135  3halfnz  9349  declei  9418  numlti  9419  faclbnd3  10722  geo2lim  11523  0.999...  11528  geoihalfsum  11529  fprodap0  11628  fprodap0f  11643  tan0  11738  cos2bnd  11767  sin4lt0  11773  eirraplem  11783  1nprm  12113  znnen  12398  tan4thpi  14232  zabsle1  14370  ex-fl  14447  trilpolemisumle  14756