Theorem eqbrtrrid 4122

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 4119	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1395   class class class wbr 4086

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 2802  df-un 3202  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087

This theorem is referenced by:  enpr1g  6967  pr2cv1  7394  endjudisj  7418  recexprlem1ssl  7846  addgt0  8621  addgegt0  8622  addgtge0  8623  addge0  8624  expge1  10831  expcnv  12058  fprodge1  12193  cos12dec  12322  3dvds  12418  bitsinv1lem  12515  ncoprmgcdne1b  12654  phicl2  12779  exmidunben  13040  prdsvalstrd  13347  znidomb  14665  sin0pilem2  15499  cosq23lt0  15550  cos0pilt1  15569  rplogcl  15596  logge0  15597  logdivlti  15598  mersenne  15714  perfectlem2  15717  lgseisen  15796  lgsquadlem1  15799