It's covered in Demailly's too-little-known book,
<a href="http://mathonline.andreaferretti.it/books/view/19/Complex-analytic-and-algebraic-geometry">Complex analytic and differential geometry,</a> though the proof given there is apparently modelled on the references you cited.

<b>Edit</b>: I just noticed that the MathOnline link currently seems to be non-functional, so here's a link to <a href="http://www-fourier.ujf-grenoble.fr/~demailly/books.html">Demailly's webpage</a>.