Theorem eqbrtrdi 4132

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 4103	. 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 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:  eqbrtrrdi  4133  pm54.43  7455  recapb  8910  nn0ledivnn  10063  xltnegi  10131  leexp1a  10919  facwordi  11065  faclbnd3  11068  resqrexlemlo  11653  efap0  12318  dvds1  12494  en1top  14888  dvef  15538  rpabscxpbnd  15751  zabsle1  15818  lgseisen  15893  lgsquadlem2  15897  upgr2wlkdc  16318  trirec0  16776