# How to prove following two statements are equivalent in Hilbert System?

statement 1: $$Γ$$ is satisfiable implies $$Γ$$ is consistent.

statement 2: If $$Γ$$ derives $$α$$ then $$Γ$$ entails $$α$$.

I can easily prove statement 1 from 2 , but not 2 from 1 (without using strong completeness theorem).

• Try: $\Gamma$ entails $\alpha$ iff $\Gamma \land \lnot \alpha$ is unsatisfiable, $\Gamma$ derives $\alpha$ iff $\Gamma \land \lnot \alpha$ is inconsistent. – Yuval Filmus Nov 25 '19 at 17:47