Vertical equivalence arrow in the middle between two lines

You need a gather* here:

\documentclass{article}

\usepackage{amsmath}

\begin{document}

\begin{gather*} 
  \forall (y_1, x_1), (y_2, x_2) \in \mathcal R_f^{-1}: y_1 = y_2 \implies x_1 = x_2 \\
                                 \Updownarrow                                        \\
    \forall (x_1, y_1), (x_2, y_2) \in \mathcal R_f: y_1 = y_2 \implies x_1 = x_2. 
\end{gather*}

\end{document}

Not exactly centered, but quite simple: Use & from the align* environment to set a center point for the alignment. The \Updownarrow then is right-aligned within the first of the two columns:

\documentclass{article}
\usepackage{amsmath}

\begin{document}

\begin{align*} 
\forall (y_1, x_1), (y_2, x_2) \in \mathcal R_f^{-1}&: y_1 = y_2 \implies x_1 = x_2 
\\ \Updownarrow \\ 
\forall (x_1, y_1), (x_2, y_2) \in \mathcal R_f&: y_1 = y_2 \implies x_1 = x_2. 
\end{align*}

\end{document}