Vertical equivalence near an itemize

Just add a column that auto-inserts these bullets.

\documentclass{article}
\usepackage{amssymb}
\usepackage{array}
\usepackage{mathtools}  
\DeclareMathOperator{\PGCD}{PGCD}
\begin{document}
$\left\Updownarrow
\begin{array}{>{\bullet}c@{\,}c}% replace \, by some other space to achieve your favorite spacing
&    d=\PGCD(a,b) \\
&    a\mathbb{Z}+b\mathbb{Z}=d\mathbb{Z}\\
&    (\forall q \in \mathbb{N}),\ (q\mid a) \wedge (q\mid b) \iff (q\mid d)
\end{array}
\right.$\bigskip

or\bigskip

$\left\Updownarrow
\begin{array}{>{\bullet}c@{\;}l}% 
&    d=\PGCD(a,b) \\
&    a\,\mathbb{Z}+b\,\mathbb{Z}=d\,\mathbb{Z}\\
&    (\forall q \in \mathbb{N}),\ (q\mid a) \wedge (q\mid b) \iff (q\mid d)
\end{array}
\right.$
\end{document}

You can add bullets in your array environment.

\documentclass{article}
\usepackage{amsmath,amssymb}
\begin{document}
\[
\left\Updownarrow
\begin{array}{rc}
    \bullet & d=PGCD(a,b) \\
    \bullet & a\mathbb{Z}+b\mathbb{Z}=d\mathbb{Z}\\
    \bullet & (\forall q \in \mathbb{N}),\ (q\mid a) \wedge (q\mid b) \iff (q\mid d)
\end{array}
\right.
\]
\end{document}