# Exists a unique

From Maths

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## Statement

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

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

- [ilmath]\exists x\in X[P(x,B)\wedge(\forall x'\in X[x'\ne x\implies ¬P(x',B)])][/ilmath]

Or using contrapositive:

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