Show that each of these statements can be used to express the fact that there is a

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Show that each of these statements can be used to express the fact that there is a unique element x such that P(x) is true.
a) ∃x∀y(P(y) ↔ x = y)
b) ∃xP(x) ∧ ∀x∀y(P(x) ∧ P(y) → x = y)
c) ∃x(P(x) ∧ ∀y(P(y) → x = y))
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