Three parameters to tune:
\eqgap
distance between double rule lines;
\overgap
separation between the text and the rules; and
\inferrulerule
thickness of two rule lines.
This MWE shows it used in math mode, but it works outside of math mode, too. I don't know enough about inference rules to know, but if the math axis is aligned with the "denominator", then this MWE would work.
\documentclass{article}
\usepackage{stackengine}
\def\eqgap{.2ex}
\def\overgap{.4ex}
\def\inferrulerule{.2pt}
\newlength\rulelength
\newlength\toplength
\newlength\bottomlength
\newcommand\myinferrule[2]{%
\stackMath%
\setlength\bottomlength{\widthof{$#1$}}%
\setlength\toplength{\widthof{$#2$}}%
\ifdim\toplength>\bottomlength%
\setlength\rulelength{\the\toplength}%
\else%
\setlength\rulelength{\the\bottomlength}%
\fi%
\mathrel{%
\stackon[\overgap]{\stackon[\eqgap]{\stackon[\overgap]{#1}%
{\rule{\the\rulelength}{\inferrulerule}}}%
{\rule{\the\rulelength}{\inferrulerule}}}{#2}%
}%
}
\begin{document}
\[
\mathrm{Math~Axis} \myinferrule{C \rightarrow C'}{C \rightarrow C_{\beta}}
\]
\end{document}

If the math axis is supposed to align with the rule, then this MWE would apply:
\documentclass{article}
\usepackage{stackengine}
\def\eqgap{.2ex}
\def\overgap{.4ex}
\def\inferrulerule{.2pt}
\newlength\rulelength
\newlength\toplength
\newlength\bottomlength
\newcommand\myinferrule[2]{%
\stackMath%
\setlength\bottomlength{\widthof{$#1$}}%
\setlength\toplength{\widthof{$#2$}}%
\ifdim\toplength>\bottomlength%
\setlength\rulelength{\the\toplength}%
\else%
\setlength\rulelength{\the\bottomlength}%
\fi%
\mathrel{%
\stackunder[\overgap]{%
\stackon[\overgap]{%
\stackanchor[\eqgap]%
{\rule{\the\rulelength}{\inferrulerule}}%
{\rule{\the\rulelength}{\inferrulerule}}%
}{#2}%
}{#1}%
}%
}
\begin{document}
\[
\mathrm{Math~Axis} \myinferrule{C \rightarrow C'}{C \rightarrow C_{\beta}}
\]
\end{document}

With these parameters, the following
\[
\myinferrule{C \rightarrow C'}{C \rightarrow C_{\beta}} ~
\myinferrule{\scriptstyle C \rightarrow C'}{\scriptstyle C \rightarrow C_{\beta}} ~
\myinferrule{\scriptscriptstyle C \rightarrow C'}{\scriptscriptstyle C \rightarrow C_{\beta}}
\]
typesets to different math styles:

If you need to regularly set this to different math styles, I can make it (at the expense of extra coding) so that the style need not be set within the arguments themselves, as it is here. So let me know if that's the case.
mathpartir
is not on CTAN, could you add a link where finding it?