My main aim is to prove or disprove that if $\Sigma \vdash \phi$ implies $\Sigma \vdash \varphi$ then $\Sigma \vdash \phi \to \varphi$ where $\Sigma$ donotes a set of sentences in propositional logic.
$\Sigma \vdash \phi$ means there is a deduction from $\Sigma$ where the deduction is a sequence $( \alpha_0 , \dots , \alpha_n)$ with $a_i$ is either in $\Sigma$ or a consequence of MP (that is, for some $j,k<i$ $\alpha_k = \alpha_j \to \alpha_i$ and $\alpha_i$ follows from them) or a tautology.
I'm completely stuck now. I tried to prove it but have no idea on how to bring $\phi$ to a deduction sequence. And I also tried to make a counterexmaple but no simple one I could find.
Even in my mere intuition, I cannot clearly judge whether it is true or false.
(I also thought of employing Completeness and Soundness Thm..)