Theorem eqbrtrrid 4080

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 2205	. 2 |- C = C
4	1, 2, 3	3brtr3g 4077	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1373   class class class wbr 4044

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045

This theorem is referenced by:  enpr1g  6890  endjudisj  7322  recexprlem1ssl  7746  addgt0  8521  addgegt0  8522  addgtge0  8523  addge0  8524  expge1  10721  expcnv  11815  fprodge1  11950  cos12dec  12079  3dvds  12175  bitsinv1lem  12272  ncoprmgcdne1b  12411  phicl2  12536  exmidunben  12797  prdsvalstrd  13103  znidomb  14420  sin0pilem2  15254  cosq23lt0  15305  cos0pilt1  15324  rplogcl  15351  logge0  15352  logdivlti  15353  mersenne  15469  perfectlem2  15472  lgseisen  15551  lgsquadlem1  15554