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**Predicate calculus and use of the form "there exists exactly one"**

Given the following utterance does the analysis necessarily follow. Is there something wrong with it or would it be deemed a correct analysis.

“Everyone has exactly one best friend”

∀x( if x is a person then there exists exactly one y such that x has a best friend y)

F(x, y) = “x has a best friend y” Pe(x) = “x is a person”

∀x(Pe(x) → ∃!y(F(x, y))

∀x(Pe(x) → ∃y(F(x, y) & ~∃z((y ≠ z) & F(x, z))

∀x(Pe(x) → ∃y(F(x, y) & ∀z((y = z) v ~F(x, z))

∀x(Pe(x) → ∃y(F(x, y) & ∀z((y = z) v ~F(x, z))

∀x(Pe(x) → ∃y(F(x, y) & ∀z((y ≠ z) → ~F(x, z)) = “Everyone has exactly one best friend”