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

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?

• If you can establish $\varphi$, then you can use $\psi$. You can also use the fact that $\varphi\to \psi$ is equivalent to $\lnot \varphi \lor\psi$. So if you can establish $\sigma$ from $\lnot \varphi$ and then also from $\psi$ (all while being able to freely use $\chi$), then you’re done. Commented May 23, 2023 at 23:45
• Just a quick note regarding my edit: notice that φ ≠ ψ ≠ χ ≠ σ does not mean that the four formulae are distinct from one another, since for example φ = χ is not precluded. Commented May 24, 2023 at 1:21

The formula $$\big((P\to Q) \land R\big) \to S$$ is equivalent to each of the following
• $$\Big((\lnot P\land R) \to S \Big) \quad\land\quad \Big((Q\land R) \to S\Big)$$
• $$R\to \Big( (\lnot P \to S ) \quad\land\quad (Q \to S) \Big).$$
• $$(R \land \lnot S) \to (P \land \lnot Q).$$
• Just to be sure: “$\rightarrow$“ here means “implies”? Commented May 24, 2023 at 2:49