Theorem eqbrtrrid 4118

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 4115	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1395   class class class wbr 4082

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 4083

This theorem is referenced by:  enpr1g  6940  pr2cv1  7356  endjudisj  7380  recexprlem1ssl  7808  addgt0  8583  addgegt0  8584  addgtge0  8585  addge0  8586  expge1  10785  expcnv  12001  fprodge1  12136  cos12dec  12265  3dvds  12361  bitsinv1lem  12458  ncoprmgcdne1b  12597  phicl2  12722  exmidunben  12983  prdsvalstrd  13290  znidomb  14607  sin0pilem2  15441  cosq23lt0  15492  cos0pilt1  15511  rplogcl  15538  logge0  15539  logdivlti  15540  mersenne  15656  perfectlem2  15659  lgseisen  15738  lgsquadlem1  15741