Theorem syl6breq 3890

Description: A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)

Hypotheses

Ref	Expression
syl6breq.1	|- ( ph -> A R B )
syl6breq.2	|- B = C

Assertion

Ref	Expression
syl6breq	|- ( ph -> A R C )

Proof of Theorem syl6breq

Step	Hyp	Ref	Expression
1		syl6breq.1	. 2 |- ( ph -> A R B )
2		eqid 2089	. 2 |- A = A
3		syl6breq.2	. 2 |- B = C
4	1, 2, 3	3brtr3g 3882	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1290   class class class wbr 3851

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-un 3004  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852

This theorem is referenced by:  syl6breqr  3891  en2eleq  6882  en2other2  6883  maxle2  10706  mertenslem2  10991  ege2le3  11022  cos01gt0  11114  sin02gt0  11115  unennn  11549