Implication Chain

Statement

Suppose that

1. $P \rightarrow Q$,
2. $Q \rightarrow R$.

Then

$$P \rightarrow R.$$

Strategy

Our goal is to prove an implication. A natural approach is to assume its antecedent, derive the consequent using the given premises, and then apply Conditional Proof.

Proof

Assume $P$.

From $P \rightarrow Q$ and $P$, by Modus Ponens, we obtain $Q$.

From $Q \rightarrow R$ and $Q$, by Modus Ponens, we obtain $R$.

Therefore,

$$P \rightarrow R,$$

by Conditional Proof.

$\square$

Remarks

This proof illustrates how Conditional Proof is often combined with inference rules such as Modus Ponens.