Theorem eqbrtri 4151

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 4137	. 2 |- ( A R C <-> B R C )
4	1, 3	mpbir 146	1 |- A R C

This proof depends on syntax axioms:   = wceq 1402   class class class wbr 4130

This proof depends on 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 proof 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 3715  df-pr 3716  df-op 3718  df-br 4131

This theorem is used by:  eqbrtrri  4153  3brtr4i  4160  exmidpw2en  7219  exmidonfinlem  7545  neg1lt0  9412  halflt1  9522  3halfnz  9743  declei  9812  numlti  9813  faclbnd3  11181  geo2lim  12283  0.999...  12288  geoihalfsum  12289  fprodap0  12388  fprodap0f  12403  tan0  12498  cos2bnd  12527  sin4lt0  12534  eirraplem  12544  1nprm  12892  ballotfilemth  13281  znnen  13289  cnfldstr  14895  tan4thpi  15942  log2tlbndlog2  16082  zabsle1  16118  ex-fl  16739  trilpolemisumle  17087