Finding a logical formula (wff) for an englsh expression

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The discussion centers on finding a logical formula that expresses the condition "there are exactly two values of x for which P(x) is true." The proposed formula is (∃x)(∃y)(∀z)(P(x) ∧ P(y) ∧ P(z) → (x=z ∨ y=z) ∧ x≠y). Participants are evaluating the validity of this formula in capturing the intended meaning. The conversation emphasizes the need for precision in logical expressions to accurately reflect the specified condition. The formula aims to ensure that only two distinct values satisfy the predicate P.
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Homework Statement



Suppose P(x) is a statement with a free variable x. Find a formula, using logical symbols, that means "there are exactly two values of x for which P(x) is true."


Homework Equations





The Attempt at a Solution




(∃x)(∃y)(∀z)(P(x) ∧ P(y) ∧ P(z) → ( x=z ∨ y=z) ∧ x≠y)

Is this valid?
 
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