[nicf]

I don't know enough about the RH examples to say what the answer is in that case. I'd be very interested in a perspective from someone who knows more than me!

In general, though, the answer to this question would depend on the specifics of the argument in question. Sometimes you might be able to salvage something; maybe there's some other setting where same methods work, or where some hypothesis analogous to the false one ends up holding, or something like that. But of course from a purely logical perspective, if I prove that P implies Q and P turns out to be false, I've learned nothing about Q.