Proof inside a proof

Question

I want to put a proof inside a proof using bussproofs. The following code produces exactly what I want.

\documentclass{amsart}
\usepackage{bussproofs}
\newcommand{\VDots}{\vphantom{\int^0}\smash[t]{\vdots}}

\begin{document}

\begin{center}
\hspace{17mm}
$\left(
        \AxiomC{$(A)$}
        \noLine
        \UnaryInfC{$\VDots$}
        \noLine
        \UnaryInfC{$B$}
    \DisplayProof
\right)$
\end{center}
\vspace{-2mm}
\begin{center}
        \AxiomC{$A\to B$}
        \AxiomC{\mbox{}}
        \noLine
        \UnaryInfC{$\VDots$}
        \noLine
        \UnaryInfC{$C$}
        \BinaryInfC{$C$}
    \DisplayProof
\end{center}

\end{document}

But obviously that's not ideal. What I want is to be able to take this chunk of code:

$\left(
        \AxiomC{$(A)$}
        \noLine
        \UnaryInfC{$\VDots$}
        \noLine
        \UnaryInfC{$B$}
    \DisplayProof
\right)$

And shove it into where I currently have a \mbox{} in the second proof. But that gets me an error.

Answer 1

It seems that busproofs does not support nesting of proofs. However, you can save the inner proof into a box and then use the box inside the outer proof.

\documentclass{amsart}
\usepackage{bussproofs}
\newcommand{\VDots}{\vphantom{\int^0}\smash[t]{\vdots}}
\begin{document}

\newsavebox\InnerProof
\sbox\InnerProof{$\left(
        \AxiomC{$(A)$}
        \noLine
        \UnaryInfC{$\VDots$}
        \noLine
        \UnaryInfC{$B$}
    \DisplayProof
\right)$%
}

\begin{center}
        \AxiomC{$A\to B$}
        \AxiomC{\usebox\InnerProof}
        \noLine
        \UnaryInfC{$\VDots$}
        \noLine
        \UnaryInfC{$C$}
        \BinaryInfC{$C$}
    \DisplayProof
\end{center}

\end{document}