Theorem eqbrtri 4114

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 4100	. 2 |- ( A R C <-> B R C )
4	1, 3	mpbir 146	1 |- A R C

Syntax hints:   = wceq 1398   class class class wbr 4093

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094

This theorem is referenced by:  eqbrtrri  4116  3brtr4i  4123  exmidpw2en  7147  exmidonfinlem  7464  neg1lt0  9310  halflt1  9420  3halfnz  9638  declei  9707  numlti  9708  faclbnd3  11068  geo2lim  12157  0.999...  12162  geoihalfsum  12163  fprodap0  12262  fprodap0f  12277  tan0  12372  cos2bnd  12401  sin4lt0  12408  eirraplem  12418  1nprm  12766  znnen  13099  cnfldstr  14654  tan4thpi  15652  zabsle1  15818  ex-fl  16439  trilpolemisumle  16770