Is This Logical Argument Valid?

[Euler2718]

Homework Statement

Determine whether the following is valid:

$$p \rightarrow \neg q , r \rightarrow q , r, \vdash \neg p$$

Homework Equations

Modus Ponens, disjunctive syllogism, double negation.

The Attempt at a Solution

I've boiled it down to

$$p \rightarrow \neg q , q, \vdash \neg p$$

However I do not understand how the book says to use disjunctive syllogism and double negation here. I've expanded the implication into its fundamental forum but I still can't see it.

 

[andrewkirk]

These problems always depend on what rules of inference, axioms and replacement rules you've been given.
$p\to \neg q$ is equivalent to $\neg p\vee \neg q$. This may be specified as a replacement rule, or as the definition of one or the other of $\to$ or $\vee$.

Using that replacement, are you able to do the problem using DNE ($\neg\neg q\equiv q$) and then DS?

 

[Euler2718]

These problems always depend on what rules of inference, axioms and replacement rules you've been given.
$p\to \neg q$ is equivalent to $\neg p\vee \neg q$. This may be specified as a replacement rule, or as the definition of one or the other of $\to$ or $\vee$.

Using that replacement, are you able to do the problem using DNE ($\neg\neg q\equiv q$) and then DS?

I've got it now, thank you. I wasn't seeing how $neg neg q \equiv q$ would fit to DS, but now it's clear.