Hypothetical Syllogism/Formulation 5/Proof 1

Proof
Let us use the following abbreviations

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

Expanding the abbreviations leads us back to:
 * $\vdash \left({q \implies r}\right) \implies \left({\left({p \implies q}\right) \implies \left({p \implies r}\right)}\right)$