PDS 70 is so far the only young disk where multiple planets have been detected by direct imaging. The disk has a large cavity when seen at submillimeter and near-infrared wavelengths, which hosts two massive planets. This makes PDS 70 the ideal target to study the physical conditions in a strongly depleted inner disk shaped by two giant planets, and in particular to test whether disk winds can play a significant role in its evolution. Using X-Shooter and HARPS spectra, we detected for the first time the wind-tracing [O i] 6300 angstrom line, and confirm the low-moderate value of mass-accretion rate in the literature. The [O i] line luminosity is high with respect to the accretion luminosity when compared to a large sample of disks with cavities in nearby star-forming regions. The FWHM and blueshifted peak of the [O i] line suggest an emission in a region very close to the star, favoring a magnetically driven wind as the origin. We also detect wind emission and high variability in the He i 10830 angstrom line, which is unusual for low accretors. We discuss that, although the cavity of PDS 70 was clearly carved out by the giant planets, the substantial inner-disk wind could also have had a significant contribution to clearing the inner disk.