center environment

% Instead of...
\begin{prooftree}
\AXC{$A \vdash A$}
\end{prooftree}

\begin{prooftree}
\AXC{$B \vdash B$}
\end{prooftree}

% ...you can use...

\begin{center}
\AXC{$A \vdash A$}
\DisplayProof
\hskip 3em
\AXC{$B \vdash B$}
\DisplayProof
\end{center}

% ...and it will be on one line.
% (Make sure you don't put a whiteline in the \begin{center} block,
% because it will be read by LaTeX as a line break
% and you still won't get the proofs side by side.)