Theorem eqbrtri 4146

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 4132	. 2 |- ( A R C <-> B R C )
4	1, 3	mpbir 146	1 |- A R C

Syntax hints:   = wceq 1402   class class class wbr 4125

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126

This theorem is referenced by:  eqbrtrri  4148  3brtr4i  4155  exmidpw2en  7209  exmidonfinlem  7535  neg1lt0  9391  halflt1  9501  3halfnz  9722  declei  9791  numlti  9792  faclbnd3  11159  geo2lim  12261  0.999...  12266  geoihalfsum  12267  fprodap0  12366  fprodap0f  12381  tan0  12476  cos2bnd  12505  sin4lt0  12512  eirraplem  12522  1nprm  12870  ballotfilemth  13259  znnen  13267  cnfldstr  14867  tan4thpi  15865  zabsle1  16032  ex-fl  16653  trilpolemisumle  16992