To prove $((\varphi \to \psi) \land \chi) \implies \sigma$, which formulas can I consider true?

Question

I'm a beginner at proof writing and I'm confused about what I can use as truth to prove a statement of the form $((\varphi \to \psi) \land \chi) \implies \sigma,$ where $\varphi, \psi, \chi$ and $\sigma$ are distinct formulas.

I'm omitting the contents of the formulas because this is something that has been confusing me for different statements.

I know for sure that I can use $\chi,$but I'm not sure what I can use from $(\varphi \to \psi)$. Can I consider the contents of $\psi$ as truth to derive $\sigma$? Can I use $\varphi$ as well?

Answer 1

The formula $$\big((P\to Q) \land R\big) \to S$$ is equivalent to each of the following

• $\Big((\lnot P\land R) \to S \Big) \quad\land\quad \Big((Q\land R) \to S\Big)$
• $R\to \Big( (\lnot P \to S ) \quad\land\quad (Q \to S) \Big).$
• $(R \land \lnot S) \to (P \land \lnot Q).$

Does that help?