Theorem eqbrtrrid 4122

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 2229	. 2 |- C = C
4	1, 2, 3	3brtr3g 4119	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1395   class class class wbr 4086

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 2802  df-un 3202  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087

This theorem is referenced by:  enpr1g  6967  pr2cv1  7391  endjudisj  7415  recexprlem1ssl  7843  addgt0  8618  addgegt0  8619  addgtge0  8620  addge0  8621  expge1  10828  expcnv  12055  fprodge1  12190  cos12dec  12319  3dvds  12415  bitsinv1lem  12512  ncoprmgcdne1b  12651  phicl2  12776  exmidunben  13037  prdsvalstrd  13344  znidomb  14662  sin0pilem2  15496  cosq23lt0  15547  cos0pilt1  15566  rplogcl  15593  logge0  15594  logdivlti  15595  mersenne  15711  perfectlem2  15714  lgseisen  15793  lgsquadlem1  15796