Hypothetical Syllogism/Formulation 1/Proof 2

Proof
This proof uses $\mathscr H_2$, Instance 2 of the Hilbert proof systems.

Recall the sequent form of the Hypothetical Syllogism:


 * $\vdash \left({q \implies r}\right) \implies \left({\left({p \implies q}\right) \implies \left({p \implies r}\right)}\right)$

Applying the Rule of Detachment $RST3$ twice, we obtain:


 * $\vdash \paren{ p \implies q } \implies \paren{ p \implies r }$

and subsequently:


 * $\vdash p \implies r$

as desired.