Existential Instantiation/Proof System

Theorem

Let $\LL$ be a specific signature for the language of predicate logic.

Let $\mathscr H$ be Hilbert proof system instance 1 for predicate logic.

Let $\map {\mathbf A} x, \mathbf B$ be WFFs of $\LL$.

Let $\FF$ be a collection of WFFs of $\LL$.

Let $c$ be an arbitrary constant symbol which is not in $\LL$.

Let $\LL'$ be the signature $\LL$ extended with the constant symbol $c$.

Suppose that we have the provable consequence (in $\LL'$):

$\FF, \map {\mathbf A} c \vdash_{\mathscr H} \mathbf B$

Then we may infer (in $\LL$):

$\FF, \exists x: \map {\mathbf A} x \vdash_{\mathscr H} \mathbf B$

Proof

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Sources

• 2009: Kenneth Kunen: The Foundations of Mathematics ... (previous) ... (next): $\text{II}.11$ Some Strategies for Constructing Proofs: Lemma $\text{II}.11.8$: Quantifier Rules (result statement)