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<p>and only if there exists a natural number  and a finite sequence of <a href="page.php?w=term_%28logic%29">term</a>s , possibly in an expansion of the <a href="page.php?w=language_%28logic%29">language</a>, with </p>

<p>
: and ,</p>

<p>such that </p>

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:</p>

<p>is valid. If it is valid, it is called a Herbrand disjunction for </p>

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:</p>

<p>Informally: a formula  in <a href="page.php?w=prenex_form">prenex form</a> containing only existential quantifiers is provable (valid) in first-order logic if and only if a disjunction composed of</p><p>
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