I've been working on a larger formula and I just wanted to double-check one section of it.
I've got the following:
$ \lnot(\forall x P(x,y) \lor R(y) )$
I've assumed that I can use the property $\lnot(P\lor Q) = \lnot P \land \lnot Q$ and $\lnot \forall x P(x) = \exists x \lnot P(x) $ to give me:
$ (\exists x \lnot P(x,y) \land \lnot R(y) )$
Would this be correct?
Your reasoning is perfectly correct!
I cannot think of anything more to add to this answer, but I must meet the character limit.