Predicate Logic Natural Deduction

Question

I'm solving the following deduction:

$$1.\forall y \ (P(y) \ \land \ \exists x \ (Q(x) \ \land \ R(y, x))) \ \ P \\2.\forall z \ (Q(z)\implies S(z) \ \land \ \exists t \ R(z, t)) \ \ P \\ \exists y \ (P(y) \ \land \ \exists x \ (S(x) \ \land \ R(y, x))) \ \ C$$

So far, I managed to get $P(y)$ through: $$3.P(y) \ \land \ \exists x \ (Q(x) \ \land \ R(y, x)) \ E\forall \ 1 \\ 4.P(y) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ E\land \ 3 $$

And from there, to get $R(y, x)$ from: $$5.\exists x \ (Q(x) \ \land \ R(y, x)) \ \ \ \ \ \ \ \ \ \ \ \ \ \ E\land \ 3 \\ 6.Q(a) \ \land \ R(y, a) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ E\exists \ 5 \\ 7.R(y, a) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ E\land \ 6$$

But I have not been able to get $S(x)$ so far. How would one proceed with that?

Answer 1

Note that you haven't used premise 2 yet. So, continue with:

\begin{array}{lll} 8. & Q(a) \to S(a) \land \exists t \ (a,t) & \forall \ E \ 2\\ 9. & Q(a) & \land \ E \ 6\\ 10. & S(a) \land \exists t \ (a,t) & \to E \ 8,9\\ 11. & S(a) & \land \ E \ 10\\ 12. & S(a) \land R(y,a) & \land \ I \ 7,11\\ 13. & \exists x (S(x) \land R(x,y)) & \exists \ I \ 12\\ 14. & P(y) \land \exists x (S(x) \land R(x,y)) & \land \ I \ 4,13\\ 15. & \exists y (P(y) \land \exists x (S(x) \land R(x,y))) & \forall \ I \ 14\\ \end{array}