Theorem eqbrtrdi 4122

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 4093	. 2 |- ( ph -> ( A R C <-> B R C ) )
4	1, 3	mpbiri 168	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1395   class class class wbr 4083

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084

This theorem is referenced by:  eqbrtrrdi  4123  pm54.43  7363  recapb  8818  nn0ledivnn  9963  xltnegi  10031  leexp1a  10816  facwordi  10962  faclbnd3  10965  resqrexlemlo  11524  efap0  12188  dvds1  12364  en1top  14751  dvef  15401  rpabscxpbnd  15614  zabsle1  15678  lgseisen  15753  lgsquadlem2  15757  trirec0  16412