Theorem eqbrtrrid 4119

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 2229	. 2 ⊢ 𝐶 = 𝐶
4	1, 2, 3	3brtr3g 4116	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1395   class class class wbr 4083

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084

This theorem is referenced by:  enpr1g  6963  pr2cv1  7384  endjudisj  7408  recexprlem1ssl  7836  addgt0  8611  addgegt0  8612  addgtge0  8613  addge0  8614  expge1  10815  expcnv  12036  fprodge1  12171  cos12dec  12300  3dvds  12396  bitsinv1lem  12493  ncoprmgcdne1b  12632  phicl2  12757  exmidunben  13018  prdsvalstrd  13325  znidomb  14643  sin0pilem2  15477  cosq23lt0  15528  cos0pilt1  15547  rplogcl  15574  logge0  15575  logdivlti  15576  mersenne  15692  perfectlem2  15695  lgseisen  15774  lgsquadlem1  15777