Theorem eqbrtrdi 4148

Description: A chained equality inference for a binary relation. (Contributed by NM, 12-Oct-1999.)

Hypotheses

Ref	Expression
eqbrtrdi.1	|- ( ph -> A = B )
eqbrtrdi.2	|- B R C

Assertion

Ref	Expression
eqbrtrdi	|- ( ph -> A R C )

Proof of Theorem eqbrtrdi

Step	Hyp	Ref	Expression
1		eqbrtrdi.2	. 2 |- B R C
2		eqbrtrdi.1	. . 3 |- ( ph -> A = B )
3	2	breq1d 4119	. 2 |- ( ph -> ( A R C <-> B R C ) )
4	1, 3	mpbiri 168	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1398   class class class wbr 4109

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 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110

This theorem is referenced by:  eqbrtrrdi  4149  pm54.43  7487  recapb  8945  nn0ledivnn  10100  xltnegi  10168  leexp1a  10956  facwordi  11102  faclbnd3  11105  resqrexlemlo  11698  efap0  12363  dvds1  12539  en1top  14942  dvef  15592  rpabscxpbnd  15805  zabsle1  15872  lgseisen  15947  lgsquadlem2  15951  upgr2wlkdc  16372  trirec0  16828