Theorem eqbrtrrid 4147

Description: B chained equality inference for a binary relation. (Contributed by NM, 17-Sep-2004.)

Hypotheses

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

Assertion

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

Proof of Theorem eqbrtrrid

Step	Hyp	Ref	Expression
1		eqbrtrrid.2	. 2 |- ( ph -> B R C )
2		eqbrtrrid.1	. 2 |- B = A
3		eqid 2234	. 2 |- C = C
4	1, 2, 3	3brtr3g 4144	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1398   class class class wbr 4111

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 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3217  df-sn 3697  df-pr 3698  df-op 3700  df-br 4112

This theorem is referenced by:  enpr1g  7040  pr2cv1  7494  endjudisj  7519  recexprlem1ssl  7950  addgt0  8724  addgegt0  8725  addgtge0  8726  addge0  8727  expge1  10942  expcnv  12194  fprodge1  12329  cos12dec  12458  3dvds  12554  bitsinv1lem  12651  ncoprmgcdne1b  12790  phicl2  12915  exmidunben  13194  prdsvalstrd  13501  znidomb  14823  sin0pilem2  15664  cosq23lt0  15715  cos0pilt1  15734  rplogcl  15761  logge0  15762  logdivlti  15763  mersenne  15882  perfectlem2  15885  lgseisen  15964  lgsquadlem1  15967