Theorem eqbrtrrid 4119

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 2229	. 2 |- C = C
4	1, 2, 3	3brtr3g 4116	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1395   class class class wbr 4083

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084

This theorem is referenced by:  enpr1g  6958  pr2cv1  7379  endjudisj  7403  recexprlem1ssl  7831  addgt0  8606  addgegt0  8607  addgtge0  8608  addge0  8609  expge1  10810  expcnv  12031  fprodge1  12166  cos12dec  12295  3dvds  12391  bitsinv1lem  12488  ncoprmgcdne1b  12627  phicl2  12752  exmidunben  13013  prdsvalstrd  13320  znidomb  14638  sin0pilem2  15472  cosq23lt0  15523  cos0pilt1  15542  rplogcl  15569  logge0  15570  logdivlti  15571  mersenne  15687  perfectlem2  15690  lgseisen  15769  lgsquadlem1  15772