Theorem eqbrtrrid 4070

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 2196	. 2 |- C = C
4	1, 2, 3	3brtr3g 4067	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1364   class class class wbr 4034

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035

This theorem is referenced by:  enpr1g  6866  endjudisj  7293  recexprlem1ssl  7717  addgt0  8492  addgegt0  8493  addgtge0  8494  addge0  8495  expge1  10685  expcnv  11686  fprodge1  11821  cos12dec  11950  3dvds  12046  bitsinv1lem  12143  ncoprmgcdne1b  12282  phicl2  12407  exmidunben  12668  prdsvalstrd  12973  znidomb  14290  sin0pilem2  15102  cosq23lt0  15153  cos0pilt1  15172  rplogcl  15199  logge0  15200  logdivlti  15201  mersenne  15317  perfectlem2  15320  lgseisen  15399  lgsquadlem1  15402