Theorem syl6eqbr 3843

Description: A chained equality inference for a binary relation. (Contributed by NM, 12-Oct-1999.)

Hypotheses

Ref	Expression
syl6eqbr.1	|- ( ph -> A = B )
syl6eqbr.2	|- B R C

Assertion

Ref	Expression
syl6eqbr	|- ( ph -> A R C )

Proof of Theorem syl6eqbr

Step	Hyp	Ref	Expression
1		syl6eqbr.2	. 2 |- B R C
2		syl6eqbr.1	. . 3 |- ( ph -> A = B )
3	2	breq1d 3816	. 2 |- ( ph -> ( A R C <-> B R C ) )
4	1, 3	mpbiri 166	1 |- ( ph -> A R C )

Syntax hints:   -> wi 4   = wceq 1285   class class class wbr 3806

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 2612  df-un 2987  df-sn 3423  df-pr 3424  df-op 3426  df-br 3807

This theorem is referenced by:  syl6eqbrr  3844  pm54.43  6554  nn0ledivnn  8955  xltnegi  9014  leexp1a  9664  facwordi  9800  faclbnd3  9803  resqrexlemlo  10084  dvds1  10445