Theorem eqbrtrrid 4025

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 2170	. 2 |- C = C
4	1, 2, 3	3brtr3g 4022	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1348   class class class wbr 3989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990

This theorem is referenced by:  enpr1g  6776  endjudisj  7187  recexprlem1ssl  7595  addgt0  8367  addgegt0  8368  addgtge0  8369  addge0  8370  expge1  10513  expcnv  11467  fprodge1  11602  cos12dec  11730  ncoprmgcdne1b  12043  phicl2  12168  exmidunben  12381  sin0pilem2  13497  cosq23lt0  13548  cos0pilt1  13567  rplogcl  13594  logge0  13595  logdivlti  13596