Prove that ($\Sigma \cup \{ \alpha \}) \vDash \beta$ if and only if $\Sigma \vDash (\alpha \to \beta)$ 
Prove that ($\Sigma \cup \{ \alpha \}) \vDash \beta$ if and only if $\Sigma \vDash (\alpha \to \beta)$

This question is quite simple, but I don't know how to express my proof formally.
I assume that there is an assignment $v$ satisfying every elenment in $\Sigma$, which also satisfies $\alpha \to \beta$, such that for any time that $\alpha$ is true, $\beta$ can't be false.
And if an assignment that satisfies $\Sigma$, and when it satisfies $\alpha$, the $\beta$ can be true.
Sorry, my description is quite unclear.
 A: $\Sigma \cup \{ \alpha \} \vDash \beta$ iff (definition $\Sigma \vDash \varphi$)
for all interpretations $I$: If $I\vDash \Sigma \cup \{ \alpha \}$ then $I\vDash \beta$ iff (definition $I \vDash \Sigma$)
for all interpretations $I$: If $I\vDash \varphi$ for all $\varphi \in \Sigma \cup \{ \alpha \}$ then $I\vDash \beta$ iff (pure logic)
for all interpretations $I$: If $I\vDash \varphi$ for all $\varphi \in \Sigma$ then if $I \vDash \alpha$ then $I\vDash \beta$ iff (semantics $\rightarrow$)
for all interpretations $I$: If $I\vDash \varphi$ for all $\varphi \in \Sigma$ then $I \vDash \alpha \rightarrow \beta$ iff (definition $I \vDash \Sigma$)
for all interpretations $I$: If $I\vDash \Sigma$ then if $I \vDash \alpha \rightarrow \beta$ iff (definition $\Sigma \vDash \varphi$)
$\Sigma \vDash \alpha \rightarrow \beta$
A: If Σ⊨(α→β), and (Σ∪{α}) holds, then both Σ and α hold (a set is only true if every member of the set holds true).  Since Σ holds, we can infer that (α→β) holds.  Since α holds also, by semantic detachment β holds.
If (Σ∪{α})⊨β and Σ holds, either α holds or α is false.  If α holds, then β follows by the first assumption also.  Since (β→(α→β)) is a logical law, (α→β) follows by detachment.  Now, ($\lnot$α→(α→β)) is a logical law also.  So, if α is false is, then $\lnot$α is true.  Thus, by semantic detachment (α→β) follows.  Both cases for α have now gotten covered.  Thus, under the hypothesis of (Σ∪{α})⊨β and Σ holding, (α→β) holds.
By the above two paragraphs, (Σ∪{α})⊨β if and only if Σ⊨(α→β).
