What does the “ (∀y)(Py → x = y)" in “∃x(Px & ∀y(Py → x = y))” mean?

Question

As known, Russell symbolise the statement ， "The present King of France is bald"， with a definite description in first order logic，"∃x(Px & ∀y(Py → x = y)&Bx)".

Why not "∃x(Px & Bx)"?

Is it just because that the present King is unique? If so, can I symbolise the statement ，"one of the leaders is bald" with "∃x(Lx & Bx)"

Answer 1

Yes, it means the present king is unique.

∃x(Px & ∀y(Py → x = y)&Bx)

"There exists an x, such that x is a present king of France, and there are no present kings of France different from x, and x is bald."