# Thread: Fol 1

1. ## Fol 1

Show that

$\{\forall x(\alpha \rightarrow \beta), \forall x \alpha \} \models \forall x \beta .$

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We show that any structure $A$ for the language and any function $s: V \rightarrow |A|$ that satisfies $\{\forall x(\alpha \rightarrow \beta), \forall x \alpha \}$ also satisfies $\forall x \beta$, where $s$ denotes a function from the set V of all variables into the universe $|A|$ of $A$. For any $d \in |A|$, we have $\models_A(\alpha \rightarrow \beta)[s(x|d)]$ and $\models_A \alpha [s(x|d)]$.
It follows that $\models_A \beta [s(x|d)]$. which in turn implies that $\models_A \forall x \beta [s]$.

Is this proof O.K or am I missing something?

Thank you.

2. ## Re: Fol 1

Yes, this proof is OK.