Answer
We can set this up in two-column format. The proof is rather long but straightforward if we go one step at a time.
$\begin{array}{ll}{\textbf { Step }} & {\textbf { Reason }} \\ {1 . \exists x \neg P(x)} & {\text { Premise }} \\ {\text { 2. } P(c)} & {\text { Existential iustantiation using }(1)} \\ {\text { 3. } \forall x(P(x) \vee Q(x))} & {\text { Premise }}\end{array}$
$\begin{array}{ll}{\text { 4. } P(c) \vee Q(c)} & {\text { Universal instantiation using }(3)} \\ {\text { 5. } Q(c)} & {\text { Disjunctive syllogism using }(4) \text { and }(2)} \\ {\text { 6. } \forall x(\neg(x) \vee S(x))} & {\text { Premise }} \\ {\text { 7. }-Q(c) \vee S(c)} & {\text { Universal instantiation using }(6)}\end{array}$
$\begin{array}{ll}{\text { 8. } S(c)} & {\text { Disjunctive syllogism using }(5) \text { and }(7), \text { since } \neg \neg Q(c) \equiv Q(c)} \\ {\text { 9. } \forall x(R(x) \rightarrow \neg S(x))} & {\text { Prenise }} \\ {\text { 10. } R(c) \rightarrow \neg S(c)} & {\text { Universal instantiation using }(9)}\end{array}$
$\begin{array}{ll}{\text { 11. } \neg R(c)} & \quad\quad\quad\quad{\text { Modus tollens using }(8) \text { and }(10), \text { since } \neg \sim S(c) \equiv S(c)} \\ {\text { 12. } \exists x \neg R(x)} & \quad\quad\quad\quad{\text { Existential generalization using }(11)}\end{array}$
Work Step by Step
We can set this up in two-column format. The proof is rather long but straightforward if we go one step at a time.
$\begin{array}{ll}{\textbf { Step }} & {\textbf { Reason }} \\ {1 . \exists x \neg P(x)} & {\text { Premise }} \\ {\text { 2. } P(c)} & {\text { Existential iustantiation using }(1)} \\ {\text { 3. } \forall x(P(x) \vee Q(x))} & {\text { Premise }}\end{array}$
$\begin{array}{ll}{\text { 4. } P(c) \vee Q(c)} & {\text { Universal instantiation using }(3)} \\ {\text { 5. } Q(c)} & {\text { Disjunctive syllogism using }(4) \text { and }(2)} \\ {\text { 6. } \forall x(\neg(x) \vee S(x))} & {\text { Premise }} \\ {\text { 7. }-Q(c) \vee S(c)} & {\text { Universal instantiation using }(6)}\end{array}$
$\begin{array}{ll}{\text { 8. } S(c)} & {\text { Disjunctive syllogism using }(5) \text { and }(7), \text { since } \neg \neg Q(c) \equiv Q(c)} \\ {\text { 9. } \forall x(R(x) \rightarrow \neg S(x))} & {\text { Prenise }} \\ {\text { 10. } R(c) \rightarrow \neg S(c)} & {\text { Universal instantiation using }(9)}\end{array}$
$\begin{array}{ll}{\text { 11. } \neg R(c)} & \quad\quad\quad\quad{\text { Modus tollens using }(8) \text { and }(10), \text { since } \neg \sim S(c) \equiv S(c)} \\ {\text { 12. } \exists x \neg R(x)} & \quad\quad\quad\quad{\text { Existential generalization using }(11)}\end{array}$
