Theorem eqbrtrrid 4124

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 4121	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1397   class class class wbr 4088

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089

This theorem is referenced by:  enpr1g  6972  pr2cv1  7400  endjudisj  7425  recexprlem1ssl  7853  addgt0  8628  addgegt0  8629  addgtge0  8630  addge0  8631  expge1  10839  expcnv  12070  fprodge1  12205  cos12dec  12334  3dvds  12430  bitsinv1lem  12527  ncoprmgcdne1b  12666  phicl2  12791  exmidunben  13052  prdsvalstrd  13359  znidomb  14678  sin0pilem2  15512  cosq23lt0  15563  cos0pilt1  15582  rplogcl  15609  logge0  15610  logdivlti  15611  mersenne  15727  perfectlem2  15730  lgseisen  15809  lgsquadlem1  15812