= Solution
Introduce the three <logical implications> from the outside in. Under assumptions $P\to Q$, $Q\to R$, and $P$, apply the <implication elimination rule> twice, then use <assumption discharge> in reverse order:
|| Line
|| Context
|| Formula
|| Justification
| 1
| $\{1\}$
| $P\to Q$
| Temporary assumption
| 2
| $\{1,2\}$
| $Q\to R$
| Temporary assumption
| 3
| $\{1,2,3\}$
| $P$
| Temporary assumption
| 4
| $\{1,2,3\}$
| $Q$
| <implication elimination rule> $\to\text{-elim}$ on 1, 3
| 5
| $\{1,2,3\}$
| $R$
| <implication elimination rule> $\to\text{-elim}$ on 2, 4
| 6
| $\{1,2\}$
| $P\to R$
| <implication introduction rule> $\to\text{-intro}$ on 5; discharge 3
| 7
| $\{1\}$
| $(Q\to R)\to(P\to R)$
| <implication introduction rule> $\to\text{-intro}$ on 6; discharge 2
| 8
| $\varnothing$
| $(P\to Q)\to((Q\to R)\to(P\to R))$
| <implication introduction rule> $\to\text{-intro}$ on 7; discharge 1
Thus
$$
\boxed{\vdash(P\to Q)\to((Q\to R)\to(P\to R)).}
$$
Back to article page
