How to deduce R from a set of sentences that I have?

Question

I can't figure out how to deduce $R$ from the set of sentences $\{R \leftrightarrow ((P \to (Q \land \lnot R)) \lor P), \, P, \, \lnot R \to Q\}$.

Any tips ? I am really bad with deductions. So far I have:

1. $P \ $ Premise
2. $\lnot R \to Q \ $ Premise
3. $R \to ((P \to (Q \land \lnot R) \lor P)) \ $ Premise
4. $\lnot R \ $ Premise
5. $\lnot\lnot R \ $ Premise
6. $R \ $ (4,5 RAA)
7. $P \to (Q \land \lnot R) \lor P \ $ (MP 3,6)

Answer 1

It is quite easy, if you consider $A \leftrightarrow B$ as a shorthand for $(A \to B) \land (B \to A)$:

1. $P \ $ Premise
2. $R \leftrightarrow ((P \to (Q \land \lnot R)) \lor P) \ $ Premise
3. $((P \to (Q \land \lnot R)) \lor P) \to R \ $ Simplification (conjunction elimination) from 2
4. $(P \to (Q \land \lnot R)) \lor P \ $ Addition (disjunction introduction) from 1
5. $R \ $ Modus ponens from 3 and 4.

Answer 2

Since $R \leftrightarrow ((P \to (Q \wedge \neg R)) \vee P)$, in order to prove $R$, it suffices to prove $(P \to (Q \wedge \neg R)) \vee P$. But this is a disjunction, so it suffices either to prove $P \to (Q \wedge \neg R)$ or to prove $P$. Can you take it from here?

Answer 3

Hint $$\begin{array}{|l}(S\vee P)\to R\\P\\\hline S\vee P\hspace{10ex} \vee\mathsf I\\R\hspace{14ex}\to\mathsf E\end{array}$$