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:


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

Applying the Rule of Detachment $\text {RST} 3$ twice, we obtain:


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

and subsequently:


 * $\vdash p \implies r$

as desired.