Formal derivation of formulae

Question

Suppose that I have a formal theory $\Gamma$ and two formulae $\phi$ and $\psi$. Suppose moreover that $\Gamma \vdash \varphi$ and $\Gamma \cup \{\varphi\}\vdash \psi$. Can we conclude that $\Gamma \vdash \psi$?

Answer 1

Yes. Start with $\Gamma$ as your givens. Since $\Gamma \vdash \phi$, you can infer $\phi$. So now you have all statements in $\Gamma \cup \phi$, and since $\Gamma \cup \phi \vdash \psi$, you can now infer $\psi$. So you can infer $\psi$ from $\Gamma$. So, $\Gamma \vdash \psi$.