Which of these formulas is a theorem of the system with axioms \(p\), \(p\to q\) and \(q\to r\), using modus ponens?

Explanation

From \(p\) and \(p\to q\), modus ponens gives \(q\); from \(q\) and \(q\to r\) it gives \(r\). The others are not derivable: they are false when every letter is true, and the axioms and the rule preserve that valuation.