Theorem eqbrtrrid 4096

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 2207	. 2 |- C = C
4	1, 2, 3	3brtr3g 4093	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1373   class class class wbr 4060

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2779  df-un 3179  df-sn 3650  df-pr 3651  df-op 3653  df-br 4061

This theorem is referenced by:  enpr1g  6915  pr2cv1  7331  endjudisj  7355  recexprlem1ssl  7783  addgt0  8558  addgegt0  8559  addgtge0  8560  addge0  8561  expge1  10760  expcnv  11976  fprodge1  12111  cos12dec  12240  3dvds  12336  bitsinv1lem  12433  ncoprmgcdne1b  12572  phicl2  12697  exmidunben  12958  prdsvalstrd  13264  znidomb  14581  sin0pilem2  15415  cosq23lt0  15466  cos0pilt1  15485  rplogcl  15512  logge0  15513  logdivlti  15514  mersenne  15630  perfectlem2  15633  lgseisen  15712  lgsquadlem1  15715