Is this proof solvable? Doubt about natural deduction

Question

I am learning logic and I found a difficult exercise. I have been exercising for three days and I start to get overwhelmed. I would like some help, to learn from my mistake... I think not to plan things well.

Premises: $P \to T, S \to \lnot R, P \wedge (Q \vee R)$

Conclusion: $T \wedge (S \to Q)$

As the conclusion is conjunction I think it may be helpful to try to get each of the individual sets separately. I got $T$, the problem is the rest.

$S \to Q$ as an implication and I think it may be helpful to start with a sub-production headed by the background of the implication and try to get as far as possible within the scope of that sub-deduction... But I feel like I don't have enough to extract things.

Thinking about what to do I got this, but I think it's not useful:

R           A
-------------
      S     A
      -------
      ¬R    E->
  ¬S  I¬
R->¬S

Q           A
-------------
      S     A
      -------
      Q    it
  S->Q
Q->(S->Q)

I appreciate the possible help, I want to move forward understanding everything possible. :( Sorry for my bad English, it's not my native language.

Answer 1

I make no attempt to duplicate the tablature, so I describe the argument.

• From $P \wedge (Q \vee R)$, obtain $Q \vee R$.
• $Q \vee R = R \vee Q$ is equivalent to $\neg R \implies Q$.
• $S \implies \neg R$ is given. By syllogism with the previous, $S \implies Q$, the clause you are seeking.