Add these to your preamble.
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}
\DeclareMathSymbol{\multimapboth}{\mathrel}{symbolsC}{"13}
Code:
\documentclass{article}
%\usepackage{txfonts}
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}
\DeclareMathSymbol{\multimapboth}{\mathrel}{symbolsC}{"13}
\begin{document}
$\multimapboth$
\end{document}

How do we know what to do?
Look in to the txfonts.sty
for the definition of \multimapboth
. It is something like
\re@DeclareMathSymbol{\multimapboth}{\mathrel}{symbolsC}{"13}
Now you need the definition of symbolsC
. Look for it in the same file.
%%%%%%%%%%%%%%%%%%
% Math Symbols C %
%%%%%%%%%%%%%%%%%%
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}
Put both together and you are good to go.
txfonts
– Steven B. Segletes May 28 '14 at 19:01