Universal Instantiation/Proof System

Theorem

Let $\map {\mathbf A} x$ be a WFF of predicate logic.

Let $\tau$ be a term which is freely substitutable for $x$ in $\mathbf A$.

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


Then:

$\forall x: \map {\mathbf A} x \vdash_{\mathscr H} \map {\mathbf A} \tau$

is a provable consequence in $\mathscr H$.


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)
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