Prove using the rules of inference that the premises ∀x (L(x)→ F(x)) and ∃x (L(x) ∧ ¬C(x)) imply the conclusion ∃x(F(x) ∧ ¬C(x)).
I'm having trouble on getting to the conclusion.
What I have so far is:
1. ∀x (L(x)→ F(x)) Hypoth
2. L(c)→ F(c) Universal instantiation
3. ∃x (L(x) ∧ ¬C(x)) Hypoth
4. L(c)∧ ~C(x) Existential instantiation
You cannot instantiate an existential to an arbitrary instance. Rather perform the existential instantiation first, then instantiate the universal to that non-arbitrary witness.
Also keep the goal in mind. You will wish to finally perform an existential generalisation to discharge some witness, so of course you will need to assume one first.
$$\begin{array}{r|l:ll} 1 & \forall x~(L(x)\to F(x)) &&\text{Hypothesis} \\ 2 & \exists x~(L(x)\wedge\neg C(x)) &&\text{Hypothesis} \\ \hline 3 & \quad L(c)\wedge \neg C(c) &2, [x\backslash c]& \text{Existential Instantiation}\\ 4 & \quad L(c)\to F(c) &1 , [x\backslash c]& \text{Universal Instantiation to witness} \\ \hline \vdots & \quad \vdots & &\text{(stuff happens)} \\ 8 & \quad F(c)\wedge\neg C(c) & & \text{(reasons)} \\ 9 & \exists x~(F(x)\wedge \neg C(x)) & 8, [c\backslash x] & \text{Existential Generalisation} \end{array}$$
Fill in the missing details.