Hypothetical Syllogism/Formulation 5/Proof 1

Proof
Let us use the following abbreviations

From Hypothetical Syllogism: Formulation 3 we have:
 * $(1): \quad \vdash \paren {\paren {p \implies q} \land \paren {q \implies r} } \implies \paren {p \implies r}$

Expanding the abbreviations leads us back to:
 * $\vdash \paren {q \implies r} \implies \paren {\paren {p \implies q} \implies \paren {p \implies r} }$