Taking the suggestion of skpblack, but limiting the importation to just those two glyphs from mathabx (since that package changes many, many glyphs), we have here denoted those two imported glyphs as \varleftrightarrow and \nvarleftrightarrow. All other math glyphs are as defined by amssymb or LaTeX, itself.
\documentclass{article}
\usepackage{amssymb}
% Setup the matha font (from mathabx.sty)
\DeclareFontFamily{U}{matha}{\hyphenchar\font45}
\DeclareFontShape{U}{matha}{m}{n}{
<5> <6> <7> <8> <9> <10> gen * matha
<10.95> matha10 <12> <14.4> <17.28> <20.74> <24.88> matha12
}{}
\DeclareSymbolFont{matha}{U}{matha}{m}{n}
% Define a subset character from that font (from mathabx.dcl)
% to completely replace the \subset character, you can replace
% \varsubset with \subset
\DeclareMathSymbol{\varleftrightarrow}{3}{matha}{"D8}
\DeclareMathSymbol{\nvarleftrightarrow}{3}{matha}{"DC}
\begin{document}
\texttt{amssymb} left-right arrow and negation
\[
A \leftrightarrow B \nleftrightarrow C
\]
\texttt{mathabx} left-right arrow and negation
\[
A \varleftrightarrow B \nvarleftrightarrow C
\]
\end{document}

mathabxdon't present this problem, but changes a little these symbols. – skpblack Aug 21 '14 at 11:05