Definition:Precisely One Function

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Let $p_1, p_2, \ldots, p_n$ be statements.

The precisely one function is the propositional function $\map P {p_1, p_2, \ldots, p_n}$ defined as:

$\map P {p_1, p_2, \ldots, p_n}$ is true if and only if precisely one of $p_1, p_2, \ldots, p_n$ is true.

$\exists p: p \in \set {p_1, p_2, \ldots, p_n}: \paren {p = \text {True} } \implies \forall x_1, x_2 \paren {x_1, x_2 \in \set {p_1, p_2, \ldots, p_n}: \paren {x_1 = \text {True} \land x_2 = \text {True} \implies \paren {x_1 = p} \land \paren {x_2 = p} } }$

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