Theorem eqbrtrrid 4095

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 2207	. 2 |- C = C
4	1, 2, 3	3brtr3g 4092	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1373   class class class wbr 4059

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060

This theorem is referenced by:  enpr1g  6913  pr2cv1  7329  endjudisj  7353  recexprlem1ssl  7781  addgt0  8556  addgegt0  8557  addgtge0  8558  addge0  8559  expge1  10758  expcnv  11930  fprodge1  12065  cos12dec  12194  3dvds  12290  bitsinv1lem  12387  ncoprmgcdne1b  12526  phicl2  12651  exmidunben  12912  prdsvalstrd  13218  znidomb  14535  sin0pilem2  15369  cosq23lt0  15420  cos0pilt1  15439  rplogcl  15466  logge0  15467  logdivlti  15468  mersenne  15584  perfectlem2  15587  lgseisen  15666  lgsquadlem1  15669