Theorem eqbrtrrid 4124

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 2231	. 2 |- C = C
4	1, 2, 3	3brtr3g 4121	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1397   class class class wbr 4088

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089

This theorem is referenced by:  enpr1g  6971  pr2cv1  7399  endjudisj  7424  recexprlem1ssl  7852  addgt0  8627  addgegt0  8628  addgtge0  8629  addge0  8630  expge1  10837  expcnv  12064  fprodge1  12199  cos12dec  12328  3dvds  12424  bitsinv1lem  12521  ncoprmgcdne1b  12660  phicl2  12785  exmidunben  13046  prdsvalstrd  13353  znidomb  14671  sin0pilem2  15505  cosq23lt0  15556  cos0pilt1  15575  rplogcl  15602  logge0  15603  logdivlti  15604  mersenne  15720  perfectlem2  15723  lgseisen  15802  lgsquadlem1  15805