Consistency of Logical Formulas has Finite Character/Proof 2

Sufficient Condition
Let $\FF$ be a consistent set of formulas.

there exists $\GG \subseteq \FF$ where $\GG$ is finite such that $\GG$ is inconsistent.

But then from Set of Logical Formulas is Inconsistent iff it has Finite Inconsistent Subset, $\FF$ is itself inconsistent.

From this contradiction it follows that $\GG$ must be consistent.

As $\GG$ is arbitrary, it follows that every finite subset of $\FF$ is consistent.

Necessary Condition
Let every finite subset of $\FF$ is consistent.

$\FF$ is inconsistent.

Then by Set of Logical Formulas is Inconsistent iff it has Finite Inconsistent Subset there exists a finite subset $\GG \subseteq \FF$ such that $\GG$ is inconsistent.

But $\GG$ is consistent.

From this contradiction it follows that $\FF$ must also be consistent.