Theorem eqbrtrrid 4018

Description: B chained equality inference for a binary relation. (Contributed by NM, 17-Sep-2004.)

Hypotheses

Ref	Expression
eqbrtrrid.1	⊢ 𝐵 = 𝐴
eqbrtrrid.2	⊢ (𝜑 → 𝐵𝑅𝐶)

Assertion

Ref	Expression
eqbrtrrid	⊢ (𝜑 → 𝐴𝑅𝐶)

Proof of Theorem eqbrtrrid

Step	Hyp	Ref	Expression
1		eqbrtrrid.2	. 2 ⊢ (𝜑 → 𝐵𝑅𝐶)
2		eqbrtrrid.1	. 2 ⊢ 𝐵 = 𝐴
3		eqid 2165	. 2 ⊢ 𝐶 = 𝐶
4	1, 2, 3	3brtr3g 4015	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1343   class class class wbr 3982

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983

This theorem is referenced by:  enpr1g  6764  endjudisj  7166  recexprlem1ssl  7574  addgt0  8346  addgegt0  8347  addgtge0  8348  addge0  8349  expge1  10492  expcnv  11445  fprodge1  11580  cos12dec  11708  ncoprmgcdne1b  12021  phicl2  12146  exmidunben  12359  sin0pilem2  13343  cosq23lt0  13394  cos0pilt1  13413  rplogcl  13440  logge0  13441  logdivlti  13442