= Solution
There is a three-line Hilbert <formal proof>. Use A1 with $\alpha=\chi$, $\beta=\psi$; then use A3 with $\alpha=\chi$, $\beta=(\psi\to\chi)$:
$$
\begin{array}{rll}
1.&\chi\to(\psi\to\chi)&\text{A1},\\
2.&(\chi\to(\psi\to\chi))\to(\neg(\psi\to\chi)\to\neg\chi)&\text{A3},\\
3.&\neg(\psi\to\chi)\to\neg\chi&\text{modus ponens on 1,2}.
\end{array}
$$
This is a direct <logical derivation> in the displayed <Hilbert proof system>. Neither the <deduction theorem for propositional logic> nor a <completeness theorem for propositional logic> is needed: the desired <logical implication> is just the contrapositive, supplied by A3, of the particular A1 instance on line 1.
Back to article page
