I am reading Model Theory by Chang and Keisler, and I am having some trouble with exercise 1.2.10, which asks me to prove that if $\Sigma \vdash \varphi$ for all $\varphi \in \Gamma$ and $\Sigma \cup \Gamma \vdash \theta$, then $\Sigma \vdash \theta$ for propositional logic. The result seems intuitively clear to me: if we can prove $\theta$ from $\Sigma$ and $\Gamma$ together, and if everything in $\Gamma$ can be proven from $\Sigma$, then $\Sigma$ itself is enough to prove $\theta$. However, I am having trouble giving a rigorous proof of this. I tried using the deduction theorem, but this only works for singleton sets, which $\Gamma$ might not be. I assume it can be proven fairly simply, but right now it escapes me.
If anyone can give a hint or a proof of this, that would be much appreciated. I would prefer to frame the proof in model-theoretic terms rather than purely logical terms.