Theorem eqbrtrrid 4129

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 2231	. 2 |- C = C
4	1, 2, 3	3brtr3g 4126	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1398   class class class wbr 4093

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094

This theorem is referenced by:  enpr1g  7015  pr2cv1  7460  endjudisj  7485  recexprlem1ssl  7913  addgt0  8687  addgegt0  8688  addgtge0  8689  addge0  8690  expge1  10901  expcnv  12145  fprodge1  12280  cos12dec  12409  3dvds  12505  bitsinv1lem  12602  ncoprmgcdne1b  12741  phicl2  12866  exmidunben  13127  prdsvalstrd  13434  znidomb  14754  sin0pilem2  15593  cosq23lt0  15644  cos0pilt1  15663  rplogcl  15690  logge0  15691  logdivlti  15692  mersenne  15811  perfectlem2  15814  lgseisen  15893  lgsquadlem1  15896