Theorem eqbrtrrid 4041

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 2177	. 2 |- C = C
4	1, 2, 3	3brtr3g 4038	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1353   class class class wbr 4005

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006

This theorem is referenced by:  enpr1g  6800  endjudisj  7211  recexprlem1ssl  7634  addgt0  8407  addgegt0  8408  addgtge0  8409  addge0  8410  expge1  10559  expcnv  11514  fprodge1  11649  cos12dec  11777  ncoprmgcdne1b  12091  phicl2  12216  exmidunben  12429  sin0pilem2  14288  cosq23lt0  14339  cos0pilt1  14358  rplogcl  14385  logge0  14386  logdivlti  14387