How can I prove it is inconsistent like this?

Question

I can't understand what the professor has taught so I don't know how to solve this problem.
Let $\mathcal L$ be a language, $\Gamma$ be a finite $\mathcal L$-theory and $\varphi$ be an $\mathcal L$-sentence. Show the following.
$\Gamma,\varphi$ is inconsistent if and only if $\Gamma \vdash\lnot\varphi$.
Would you please tell me how to solve it?

Answer 1

Hint

We have to start from the definition of inconsistent set of formulas: $Γ$ is inconsistent iff $Γ ⊢ \varphi$ and $Γ ⊢ ¬ \varphi$, for some formula $\varphi$.

For the "simple" part: if $\Gamma \vdash \lnot \varphi$, then also $\Gamma, \varphi \vdash \lnot \varphi$.

But obviously $\Gamma, \varphi \vdash \varphi$, and thus $\Gamma, \varphi$ is inconsistent.

For the othe part: if $\Gamma, \varphi$ is inconsistent, then $\Gamma, \varphi \vdash \lnot \psi$ and $\Gamma, \varphi \vdash \lnot \psi$, for some $\psi$.

Apply Negation Introduction rule to conclude with: $\Gamma \vdash \lnot \varphi$.