I read that the soundness theorem states that if $\Gamma \vdash \phi$, then $\Gamma \models \phi$, where $\Gamma \vdash \phi$ means that $\phi$ is a deductive consequence of $\Gamma$ under some first-order deductive calculus, and $\Gamma \models \phi$ means that $\Gamma$ logically implies $\phi$.
I read somewhere that the soundness theorem is actually equivalent to the statement:
- if $\Gamma$ is satisfiable, then $\Gamma$ is consistent
where $\Gamma$ is satisfiable if there is some structure $\mathfrak{U}$ and assignment function $s$ such that $\mathfrak{U}$ satisfies every member of $\Gamma$ with $s$. How is the statement if $\Gamma$ is satisfiable, then $\Gamma$ is consistent equivalent to the soundness theorem ?