The Deduction Theorem states: $\Sigma \cup \theta \vdash \phi <=> \Sigma \vdash \theta \to \phi$ and I will claim to prove it as follows:
<-: Assume $\Sigma \vdash \theta \to \phi$. Because of monotony it follows that $\Sigma \cup \theta \vdash \theta \to \phi $ and because of triviality we also have $\Sigma \cup \theta \vdash \theta$ which by mp leads us to $\Sigma \cup \theta \vdash \phi$.
->: Assume $\Sigma \cup \theta \vdash \phi$ and assume soundness, so we know that $\Sigma \cup \theta \vdash \phi$ is a tautology of the form $ (\Sigma \lor \theta) \to \phi$ and $\Sigma \vdash \theta \to \phi$ is a tautology of the form $\Sigma \to (\theta \to \phi)$. Now we assume the implication to be false, i.e. $ ((\Sigma \lor \theta) \to \phi) \land \lnot(\Sigma \to (\theta \to \phi))$ to be true. A truth table shows us that this conjunction is always false which means that the assumption of the implication to be false was false itself which makes the implication true. $ \square$
Is this a legit proof or where does it fail? I am especially curious if I can go from $\Sigma \cup \theta$ to $\Sigma \lor \theta$ and if not why not. If my proof fails can you rescue it somehow because I like my proof since it's simple while most proofs of the deduction theorem look rather complicated, mostly they prove it by induction, probably because they don't assume soundness like I do?
Long comment
The usual proof of the Deduction Theorem is "boring" exactly because it does not assume soundness: it is a purely syntactical proof.
A semantic one amounts to showing: if $A \vDash B$, then $\vDash A \to B$, which is straightforward (for propositional logic it is a simple check with truth table).
The proof of DT is by induction and this is the reason why it is boring. We can find it in every textbook that uses a so-called Hilbert-style proof system (like e.g. Mendelson, page 30), i.e. a proof system with axioms and Modus Ponens rule of inference: the lesser is the number of axioms, the shortest the proof.
It is worth noting that there are proof systems, like e.g. Natural Deduction where the DT is built-in into the rules.
Two points must be stressed:
(i) the DT holds also when $Σ$ is infinite, while the tautology $Σ → (θ → \phi)$ is a well-formed formula only when it is a finite expression.
(ii) the inductive proof of the DT gives a recipe (a "mechanical" procedure) to produce, form the derivation $Σ,θ \vdash \phi$ the new derivation $Σ ⊢ θ → \phi$.