Theorem syl6eqbrr 3831

Description: A chained equality inference for a binary relation. (Contributed by NM, 4-Jan-2006.)

Hypotheses

Ref	Expression
syl6eqbrr.1	|- ( ph -> B = A )
syl6eqbrr.2	|- B R C

Assertion

Ref	Expression
syl6eqbrr	|- ( ph -> A R C )

Proof of Theorem syl6eqbrr

Step	Hyp	Ref	Expression
1		syl6eqbrr.1	. . 3 |- ( ph -> B = A )
2	1	eqcomd 2087	. 2 |- ( ph -> A = B )
3		syl6eqbrr.2	. 2 |- B R C
4	2, 3	syl6eqbr 3830	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1285   class class class wbr 3793

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 2064

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794

This theorem is referenced by: (None)