Theorem eqbrtrrid 4150

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 2234	. 2 |- C = C
4	1, 2, 3	3brtr3g 4147	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1398   class class class wbr 4114

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 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115

This theorem is referenced by:  enpr1g  7051  pr2cv1  7505  endjudisj  7530  recexprlem1ssl  7964  addgt0  8739  addgegt0  8740  addgtge0  8741  addge0  8742  expge1  10962  expcnv  12215  fprodge1  12350  cos12dec  12479  3dvds  12575  bitsinv1lem  12672  ncoprmgcdne1b  12811  phicl2  12936  ballotfilemfrcn0  13217  exmidunben  13261  prdsvalstrd  13563  znidomb  14932  sin0pilem2  15773  cosq23lt0  15824  cos0pilt1  15843  rplogcl  15870  logge0  15871  logdivlti  15872  mersenne  15991  perfectlem2  15994  lgseisen  16073  lgsquadlem1  16076