I cannot find a name for this metatheorem. By reverse deduction theorem I mean $\Gamma \vdash \phi \to \psi$ implies $\Gamma, \phi \vdash \psi$. There is another question regarding this but for propositional logic.
I am asked to prove $\Gamma \vdash \phi \to \psi$ implies $\Gamma, \phi \vdash \psi$ when $\Gamma, \phi, \psi$ are all sentences. I have to prove this given the following rules:
Now, my proof is as follows:
We can derive $\Gamma, \phi \vdash \phi \to \psi$ by the thinning rule. We also have $\Gamma, \phi \vdash \phi$ by assumption. Then by modus ponens $\Gamma, \phi \vdash \psi$.
But, to me it seems that this proof should work when dropping the restriction of sentences. So why did the exercise require that they be sentences? Am I missing something?