I'm currently working on this problem:
“All movies produced by John Sayles are wonderful. John Sayles produced a movie about coal miners. Therefore, there is a wonderful movie about coal miners.”
My domain/Universe consists of all movies.
P(x,y): x produced y
W(x): x is a wonderful movie
c : a movie about coal miners
Premises 1. $∀x(P (John Sayles, x) → W(x))$
Premises 2. $P(John Sayles, c)$
Therefore 3. $∃xW(c)$
I want to take premises 1 and use Universal Instantiation to get this: $P(John Sayles, c) → W(c))$
I'm not sure if that's an ok thing to do with universal instantiation or if it's wrong.
Yes, when you instantiate you have to instantiate every free variable quantified by the quantifier that you are dropping.
And that is exactly what you do when going from $\forall x (P(JohnSayles,x) \to W(x))$ to $P(JohnSayles,c) \to W(c)$, so what you do here is correct.
What is not correct, however, is your symbolization. Note that your conclusion is really weird: $\exists x W(c)$ has no free variables $x$, so what are you quantifying here?
So, I would use a predicate $C(x)$ that means that '$x$ is a movie about coal miners'. With that, you get:
$\forall x (P(j,x) \to W(x))$
$\exists x (P(j,x) \land C(x))$
$\therefore \exists x (W(x) \land C(x))$