We present Lya and UV-nebular emission line properties of bright Lya emitters (LAEs) at z=6-7 with a luminosity of log L_Lya/[erg s-1] = 43-44 identified in the 21-deg2 area of the SILVERRUSH early sample developed with the Subaru Hyper Suprime-Cam (HSC) survey data. Our optical spectroscopy newly confirm 21 bright LAEs with clear Lya emission, and contribute to make a spectroscopic sample of 96 LAEs at z=6-7 in SILVERRUSH. From the spectroscopic sample, we select 7 remarkable LAEs as bright as Himiko and CR7 objects, and perform deep Keck/MOSFIRE and Subaru/nuMOIRCS near-infrared spectroscopy reaching the 3sigma-flux limit of ~ 2x10^{-18} erg s-1 for the UV-nebular emission lines of He II1640, C IV1548,1550, and O III]1661,1666. 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 (EW_0) of ~2-4 A for He II, C IV, and O III] lines. Here we also investigate the VLT/X-SHOOTER spectrum of CR7 whose 6 sigma detection of He II is claimed by Sobral et al. Although two individuals and the ESO-archive service carefully re-analyze 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. 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 EW_0 and UV-continuum luminosity, which are similar to those found at z~2-3.
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