Theorem eqbrtrrid 4161

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 2238	. 2 |- C = C
4	1, 2, 3	3brtr3g 4158	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1402   class class class wbr 4125

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126

This theorem is referenced by:  enpr1g  7075  pr2cv1  7531  endjudisj  7556  recexprlem1ssl  7990  addgt0  8766  addgegt0  8767  addgtge0  8768  addge0  8769  expge1  10991  expcnv  12249  fprodge1  12384  cos12dec  12513  3dvds  12609  bitsinv1lem  12706  ncoprmgcdne1b  12845  phicl2  12970  ballotfilemfrcn0  13251  exmidunben  13295  prdsvalstrd  13597  znidomb  14965  sin0pilem2  15806  cosq23lt0  15857  cos0pilt1  15876  rplogcl  15903  logge0  15904  logdivlti  15905  mersenne  16025  perfectlem2  16028  lgseisen  16107  lgsquadlem1  16110