How to add a short slash (line segment) on one of the \leftrightarrows?

Question

I want to know how to get two paralleled reverse single arrows between proposition A and B, and one of the arrows with a short slash, which means that A can deduces B but B cannot deduce A. Shortly, how to add a short slash (line segment) on one of the \leftrightarrows?

\documentclass{article}
\usepackage{amsmath,amssymb}

\begin{document}

Proposition A$\leftrightarrows$Proposition B

I want to add a slash (line segment) on the leftarrow, 
it means that A can deduce to B but B cannot deduce to A.

\end{document}

Answer 1

A possibility combining \nleftarrow from amssymb with the old style \rightarrow provided by old-arrows:

\documentclass{article}

\usepackage{amsmath}
\usepackage{amssymb}
\usepackage[old]{old-arrows}

\makeatletter
\newcommand{\notleftyesrightarrow}{\mathrel{\mathpalette\notleft@yesrightarrow\relax}}
\newcommand{\notleft@yesrightarrow}[2]{%
   \vcenter{%
      \m@th
      \offinterlineskip
      \hbox{$#1\nleftarrow$}
      \hbox{$#1\varrightarrow$}
   }%
}
\makeatother
\begin{document}

$A \notleftyesrightarrow B \leftrightarrows C$
$\scriptstyle A \notleftyesrightarrow B \leftrightarrows C$
$\scriptscriptstyle A \notleftyesrightarrow B \leftrightarrows C$

\end{document}

Loading the package old-arrows with the option old does not replace the standard arrows but rather provides the onld-style ones with var prepended to the name (here \varrightarrow).

Answer 2

\documentclass{article}
\usepackage{centernot}

\begin{document}
\[
    A \leftrightarrow
    \begin{array}{c}
        \rightarrow\\[-4pt]
        \centernot{\leftarrow}
    \end{array}
    B
\]
\end{document}

Use the centernot package for the slash and an array environment to have one arrow above the other, decrease the vertical spacing by 4pt using \\[-4pt] instead of just \\