© The Author 2017. Published by Oxford University Press on behalf of the Astronomical Society of Japan. All rights reserved. We present Lyα and UV-nebular emission line properties of bright Lyα emitters (LAEs) at z = 6-7 with a luminosity of log LLyα/[erg s-1] = 43-44 identified in the 21 deg2 area of the SILVERRUSH early sample developed with the Subaru Hyper Suprime-Cam survey data. Our optical spectroscopy newly confirms 21 bright LAEs with clear Lyα emission, and contributes to making a spectroscopic sample of 96 LAEs at z = 6-7 in SILVERRUSH. From the spectroscopic sample, we select seven remarkable LAEs as bright as Himiko and CR7 objects, and perform deep Keck/MOSFIRE and Subaru/nuMOIRCS near-infrared spectroscopy reaching the 3 σ flux limit of ∼2 × 10-18 erg s-1 for the UV-nebular emission lines of He II λ1640, CIV λλ1548,1550, and OIII]λλ1661,1666. Except for one tentative detection of C IV, we find no strong UV-nebular lines down to the flux limit, placing the upper limits of the rest-frame equivalent widths (EW0) of ∼2-4 A for CIV, He II, and OIII] lines. We also investigate the VLT/X-SHOOTER spectrum of CR7 whose 6 σ detection of He II is claimed by Sobral et al. Although two individuals and the ESO archive service carefully reanalyzed the X-SHOOTER data that are used in the study of Sobral et al., no He II signal of CR7 is detected, supportive of weak UV-nebular lines of the bright LAEs even for CR7. The spectral properties of these bright LAEs are thus clearly different from those of faint dropouts at z ∼ 7 that have strong UV-nebular lines shown in the various studies. Comparing these bright LAEs and the faint dropouts, we find anti-correlations between the UV-nebular line EW0 and the UV-continuum luminosity, which are similar to those found at z ∼ 2-3.