apter 2 introduces the notation <span class="math inline">\(\exists ! x\,P(x)\)</span> to mean “there is exactly one <span class="math inline">\(x\)</span> such that <span class="math inline">\(P(x).\)</span>” This can be thought of as an abbreviation for <span class="math inline">\(\exists x(P(x) \wedge \neg\exists y(P(y) \wedge y \ne x))\)</span>. By the quantifier negation, De Morgan, and conditional laws, this is equivalent to <span class="math inline">\(\exists x(P(x) \wedge \forall y(P(y) \to y = 