Exists a unique

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Created just to note down my work! Flesh out later

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Statement

• Suppose we have $\exists x\in X[P(x,B)]$ where $P$ is some formula (FOL) and $B$ is some baggage (like more variables from the rest of the statement this might be in)

To extend this to there exists a unique $x$ such that {M|P(x,B)}} we can write:

• $\exists x\in X[P(x,B)\wedge(\forall x'\in X[x'\ne x\implies ¬P(x',B)])]$

Or using contrapositive:

• $\exists x\in X[P(x,B)\wedge(\forall x'\in X[P(x',B)\implies x'=x])]$ (confirmed by third party)