Hilbert System Logical Axiom 1 follows from Axioms 2 and 3

Hint: Remember you get to pick what $\psi$ is. Is there any formula $\psi$ such that you know $(\phi\to\psi)$ is true?

More details are hidden below.

By axiom 2, if you choose $\psi$ to have the form $(\psi'\to\phi)$ for some formula $\psi'$, then you know $(\phi\to \psi)$ is true.

By (3) we have $$ (\phi\to((\phi\to\phi)\to\phi))\to((\phi\to(\phi\to\phi))\to(\phi\to\phi)) $$ By (2), $$ \phi\to ((\phi\to\phi)\to\phi) $$ So $$ ((\phi\to(\phi\to\phi))\to(\phi\to\phi)) $$ by Modus Ponens.

By (2), $$ \phi\to (\phi\to\phi) $$ so $$ \phi\to\phi $$ by Modus Ponens again.