Theorem eqbrtrdi 4083

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 4054	. 2 |- ( ph -> ( A R C <-> B R C ) )
4	1, 3	mpbiri 168	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1373   class class class wbr 4044

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045

This theorem is referenced by:  eqbrtrrdi  4084  pm54.43  7298  recapb  8744  nn0ledivnn  9889  xltnegi  9957  leexp1a  10739  facwordi  10885  faclbnd3  10888  resqrexlemlo  11324  efap0  11988  dvds1  12164  en1top  14549  dvef  15199  rpabscxpbnd  15412  zabsle1  15476  lgseisen  15551  lgsquadlem2  15555  trirec0  15983