Theorem eqbrtrrid 4034

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 2175	. 2 |- C = C
4	1, 2, 3	3brtr3g 4031	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1353   class class class wbr 3998

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737  df-un 3131  df-sn 3595  df-pr 3596  df-op 3598  df-br 3999

This theorem is referenced by:  enpr1g  6788  endjudisj  7199  recexprlem1ssl  7607  addgt0  8379  addgegt0  8380  addgtge0  8381  addge0  8382  expge1  10525  expcnv  11478  fprodge1  11613  cos12dec  11741  ncoprmgcdne1b  12054  phicl2  12179  exmidunben  12392  sin0pilem2  13772  cosq23lt0  13823  cos0pilt1  13842  rplogcl  13869  logge0  13870  logdivlti  13871