In Theorem C.2.2.13 of Johnstone's Elephant, it is asserted that any locally small cocomplete quasitopos with a strong generator (a "generating set" in Johnstone's terminology) is locally presentable (among other things). However, the proof of this specific implication seems lacking to me; Johnstone simply says that one can deduce local presentability from these hypotheses by arguing as in D.2.3.7, which asserts that any Grothendieck topos is locally presentable. I do not obviously see how one can modify the proof of this latter result to show that any locally small cocomplete quasitopos with a strong generator is locally presentable, unless one already knew that such a category is a Grothendieck quasitopos, but this implication is not (explicitly) proved in Theorem C.2.2.13 either.
So my question is, given that Johnstone does not seem to give a (direct) proof of the claim that any locally small cocomplete quasitopos with a strong generator is locally presentable, how could one establish this?