Select all that apply

To prove that there is a unique element x such that P$($x$)$ is true is the same as showing which of these statements is true?

Multiple select question.

$$x $($P$($x$)$y$($y $!=$ x $->$P$($y$)))$

$$x $($P$($x$)$y$($P$($y$)->$ x $=$ y$))$

$$x $($P$($x$)$y$($P$($x$)->$ x $=$ y$))$

$$x $($P$($x$)$y$($y $!=$ x $->$P$($y$)))$

$$x $($P$($x$)$y$($y $!=$ x $->$P$($y$)))$

$$x $($P$($x$)$y$($P$($x$)->$ x $=$ y$))$

$$x $($P$($x$)$y$($y $!=$ x $->$P$($y$)))$