It's covered in Demailly's too-little-known book, Complex analytic and differential geometry, though the proof given there is apparently modelled on the references you cited.

It's covered in Demailly's too-little-known book, Complex analytic and differential geometry, though the proof given there is apparently modelled on the references you cited.

Edit: I just noticed that the MathOnline link currently seems to be non-functional, so here's a link to Demailly's webpage.