Theorem eqbrtrrid 4069

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 2196	. 2 |- C = C
4	1, 2, 3	3brtr3g 4066	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1364   class class class wbr 4033

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034

This theorem is referenced by:  enpr1g  6857  endjudisj  7277  recexprlem1ssl  7700  addgt0  8475  addgegt0  8476  addgtge0  8477  addge0  8478  expge1  10668  expcnv  11669  fprodge1  11804  cos12dec  11933  3dvds  12029  ncoprmgcdne1b  12257  phicl2  12382  exmidunben  12643  znidomb  14214  sin0pilem2  15018  cosq23lt0  15069  cos0pilt1  15088  rplogcl  15115  logge0  15116  logdivlti  15117  mersenne  15233  perfectlem2  15236  lgseisen  15315  lgsquadlem1  15318