  In this note we study several topics related to the schema of local
reflection $\mathsf{Rfn}(T)$ and its partial and relativized variants. Firstly,
we introduce the principle of uniform reflection with $\Sigma_n$-definable
parameters, establish its relationship with the relativized local reflection
principles and corresponding versions of induction with definable parameters.
Using this schema we give a new model-theoretic proof of the
$\Sigma_{n+2}$-conservativity of uniform $\Sigma_{n+1}$-reflection over
relativized local $\Sigma_{n+1}$-reflection. We also study the proof-theoretic
strength of Feferman's theorem, i.e., the assertion of $1$-provability in $S$
of the local reflection schema $\mathsf{Rfn}(S)$, and its generalized versions.
We relate this assertion to the uniform $\Sigma_2$-reflection schema and, in
particular, obtain an alternative axiomatization of $\mathsf{I}\Sigma_1$.
