Does second-order arithmetic (Z2) prove soundness and uniform reflection for first-order arithmetic (PA)? (1)  Does full second-order arithmetic (Z2) prove soundness and uniform reflection schemas for first-order arithmetic (PA)?  That is, do we have for all formulas $\phi$:
$$ \underset \phi \forall \; Z2 \vdash \square_{PA} \ulcorner \phi \urcorner \rightarrow \phi $$
and
$$ \underset \phi \forall \; Z2 \vdash \forall x: \big ( \square_{PA} \ulcorner \phi(x) \urcorner \rightarrow \phi(x) \big ) $$
?
(2)  What would be a good reference to cite to this effect?
 A: Yes; if $\phi(x)$ is a formula of first-order arithmetic, then $Z_2$ proves
$$
(\forall x)[ \Box_{PA}\ulcorner \phi(x)\urcorner \Rightarrow \phi(x) ]
$$
(The rest of this answer would go through with $Z_2$ replaced by the subsystem with only $\Pi^1_1$ comprehension, and likely in even weaker subsystems, although that goes beyond the question.) 
The argument is sketched as follows:


*

*$Z_2$ can construct a truth function $T_1$ for all sentences of first-order arithmetic with parameters in $\mathbb{N}$.

*$Z_2$ can prove that each axiom of PA is true under this truth function.

*$Z_2$ can prove that the set of true sentences is closed under the inference rules of PA. In other words $Z_2$ can formalize and prove $\mathbb{N} \vDash \text{PA}$.

*So $Z_2$ proves, for each $\phi(x)$ in the language of PA, that
$$(\forall x)[\Box_{PA}\ulcorner \phi(x)\urcorner \Rightarrow T_1(\ulcorner \phi(x)\urcorner) = 1]$$
(In fact $Z_2$ proves this with a quantifier over codes for PA-formulas $\phi$, but this isn't needed for the argument at hand.)

*For each particular formula $\phi$ of PA, $Z_2$ proves 
$$(\forall x)[ T_1(\ulcorner \phi(x)\urcorner) = 1
\Rightarrow \phi(x)].$$
The proof is by induction on the structure of $\phi$ using the truth definition that was used to define $T_1$.

*The rest of the proof is immediate.
As for a reference, this is nothing deep, it is just an exercise.  I don't know whether it is mentioned anywhere in the literature, but the techniques are all standard. 
