Theorem eqbrtrrid 4081

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 4078	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1373   class class class wbr 4045

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 4046

This theorem is referenced by:  enpr1g  6892  endjudisj  7324  recexprlem1ssl  7748  addgt0  8523  addgegt0  8524  addgtge0  8525  addge0  8526  expge1  10723  expcnv  11848  fprodge1  11983  cos12dec  12112  3dvds  12208  bitsinv1lem  12305  ncoprmgcdne1b  12444  phicl2  12569  exmidunben  12830  prdsvalstrd  13136  znidomb  14453  sin0pilem2  15287  cosq23lt0  15338  cos0pilt1  15357  rplogcl  15384  logge0  15385  logdivlti  15386  mersenne  15502  perfectlem2  15505  lgseisen  15584  lgsquadlem1  15587