I'd be careful about going that small.
It's not that a 51 couldn't fit a 5'9" person, it's just that it seems likely that it's on the small side. I'm not doing a blind internet fit thing here: by all means, if you know that this size works for you than it does. And if somebody who knows what's up, and has been able to see you on this bike (maybe before or after taking measurements) confirms this fit, then I'm not sure I have a real argument. But (a) I'd get that inseam measurement checked by somebody else, to hard contact, and (b) I'd wonder about more than vertical fit, even if the inseam really is that short. Fitting only on the basis of one's inseam can create all sorts of problems. For one thing, focusing on an idealized seat tube can point you to a screwy top tube length (especially--but not only--if your proportions are a bit off the norm). Off by a cm or even two? You might still be perfectly happy with the bike's handling if you change the stem length by a cm or two. Off by four or five? Much more likely to be a problem.
I'd recommend finding some good, local, in-person fit advice. Also, if the current bike seems like a bit of a stretch, why not just try a stem that's shorter by a cm or two before chucking the bike?