Did I solve a basic derivation problem correctly?

Question

The following problem is from Mathematical Logic by Ian Chiswell and Wilfrid Hodges, 2007.

Answer 1

Not exactly...

I think that, due to the fact that the assumption $\varphi$ has not been "crossed away", it is correct to consider it undischarged.

But $\psi$ is not listed explicitly as an assumption; thus, I think that the correct answer is :

$\{ \varphi \} \vdash (\varphi \to (\psi \to \varphi))$.

The following :

$$\dfrac{ \varphi }{ \psi \to \varphi }$$

is a correct application of the rule ($\to$-I); "semantically", if $\varphi$ is true, then $\psi \to \varphi$ is true, for $\psi$ whatever.