First Order Logic: True in a closed set of formulas

Question

Need some help with this problem:

If $\Gamma \cup \{ \varphi \}$ is a set of closed formulas, then $\Gamma \cup \{ \varphi \} \vDash \psi$ iff $\Gamma \vDash \varphi \rightarrow \psi$

My attempt: ($\Rightarrow$) For all model $\mathfrak{A}$, if $\mathfrak{A} \vDash \Gamma \cup \{ \varphi \}$, then $\mathfrak{A} \vDash \psi$. As $\mathfrak{A} \vDash \Gamma \cup \{ \varphi \}$, and $\Gamma \cup \{ \varphi \}$ is a closed set of formulas, then $\mathfrak{A} \vDash \Gamma$ or $\mathfrak{A} \vDash \varphi$. Thus, $\mathfrak{A} \vDash \varphi \rightarrow \psi$, and then, $\Gamma \vDash \varphi \rightarrow \psi$

($\Leftarrow$) Don't know how start.

Thanks for your help!

Answer 1

Let $\mathfrak{A}\vDash \Gamma$ if $\mathfrak{A}\vDash\neg\varphi$ then $\mathfrak{A}\vDash\varphi\rightarrow\psi$. If on the other hand $\mathfrak{A}\vDash\varphi$ then $\mathfrak{A}\vDash \Gamma\cup\{\varphi\}$ so we have $\mathfrak{A}\vDash\psi$ and we then have $\mathfrak{A}\vDash\varphi\rightarrow \psi$.