Theorem eqbrtrrid 4129

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 2231	. 2 ⊢ 𝐶 = 𝐶
4	1, 2, 3	3brtr3g 4126	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1398   class class class wbr 4093

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094

This theorem is referenced by:  enpr1g  7015  pr2cv1  7443  endjudisj  7468  recexprlem1ssl  7896  addgt0  8671  addgegt0  8672  addgtge0  8673  addge0  8674  expge1  10882  expcnv  12126  fprodge1  12261  cos12dec  12390  3dvds  12486  bitsinv1lem  12583  ncoprmgcdne1b  12722  phicl2  12847  exmidunben  13108  prdsvalstrd  13415  znidomb  14734  sin0pilem2  15573  cosq23lt0  15624  cos0pilt1  15643  rplogcl  15670  logge0  15671  logdivlti  15672  mersenne  15791  perfectlem2  15794  lgseisen  15873  lgsquadlem1  15876