Direct Proof with more than one conditional.

Question

If I have a statement such as $p \implies (q \implies r)$ that I want to prove it with a direct proof. Would I assume $p \land q$ are true and then follow the axioms and previously proven theorems to reach $r$? Does that mean that the statement only holds when $p \land q$ are true?

Answer 1

For the first part of the question your are correct.

For the second:

Does that mean that the statement only holds when $p∧q$ are true?

No, if $p$ is false then $p\Rightarrow(q\Rightarrow r)$ also holds because false implies anything. Similarly if $q$ is false, then $q\Rightarrow r$ is true and thus $p\Rightarrow(q\Rightarrow r)$ also holds (anything implies the truth).