Theorem syl5eqbrr 3845

Description: B chained equality inference for a binary relation. (Contributed by NM, 17-Sep-2004.)

Hypotheses

Ref	Expression
syl5eqbrr.1	⊢ 𝐵 = 𝐴
syl5eqbrr.2	⊢ (𝜑 → 𝐵𝑅𝐶)

Assertion

Ref	Expression
syl5eqbrr	⊢ (𝜑 → 𝐴𝑅𝐶)

Proof of Theorem syl5eqbrr

Step	Hyp	Ref	Expression
1		syl5eqbrr.2	. 2 ⊢ (𝜑 → 𝐵𝑅𝐶)
2		syl5eqbrr.1	. 2 ⊢ 𝐵 = 𝐴
3		eqid 2083	. 2 ⊢ 𝐶 = 𝐶
4	1, 2, 3	3brtr3g 3842	1 ⊢ (𝜑 → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1285   class class class wbr 3811

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-un 2988  df-sn 3428  df-pr 3429  df-op 3431  df-br 3812

This theorem is referenced by:  enpr1g  6445  recexprlem1ssl  7095  addgt0  7829  addgegt0  7830  addgtge0  7831  addge0  7832  expge1  9829  ncoprmgcdne1b  10851  phicl2  10970