Theorem eqbrtrrid 4164

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 2238	. 2 ⊢ 𝐶 = 𝐶
4	1, 2, 3	3brtr3g 4161	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1402   class class class wbr 4128

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3714  df-pr 3715  df-op 3717  df-br 4129

This theorem is referenced by:  enpr1g  7079  pr2cv1  7535  endjudisj  7560  recexprlem1ssl  7994  addgt0  8770  addgegt0  8771  addgtge0  8772  addge0  8773  expge1  10996  expcnv  12254  fprodge1  12389  cos12dec  12518  3dvds  12614  bitsinv1lem  12711  ncoprmgcdne1b  12850  phicl2  12975  ballotfilemfrcn0  13256  exmidunben  13300  prdsvalstrd  13603  znidomb  14976  sin0pilem2  15866  cosq23lt0  15917  cos0pilt1  15936  rplogcl  15963  logge0  15964  logdivlti  15965  mersenne  16094  perfectlem2  16097  lgseisen  16176  lgsquadlem1  16179