See here a stronger one: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=51&t=200683

\begin{align*} \hfill\\ \left. \begin{gathered} 2(1-a)(1-c) = (a+c-1)^2 + b^2 + d^2 \geq b^2 + d^2 \\ 2(1-b)(1-d) = (b+d-1)^2 + a^2 + c^2 \geq a^2 + c^2 \hfill\\ \end{gathered} \right \} \implies \prod_{cyc} (1-a) \geq \frac{1}{4} \sum_{cyc} a^2b^2 \geq abcd. \end{align*}