Theorem eqbrtrrid 4066

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 2193	. 2 |- C = C
4	1, 2, 3	3brtr3g 4063	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1364   class class class wbr 4030

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3158  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031

This theorem is referenced by:  enpr1g  6854  endjudisj  7272  recexprlem1ssl  7695  addgt0  8469  addgegt0  8470  addgtge0  8471  addge0  8472  expge1  10650  expcnv  11650  fprodge1  11785  cos12dec  11914  ncoprmgcdne1b  12230  phicl2  12355  exmidunben  12586  znidomb  14157  sin0pilem2  14958  cosq23lt0  15009  cos0pilt1  15028  rplogcl  15055  logge0  15056  logdivlti  15057  lgseisen  15231  lgsquadlem1  15234