"Implied atomic propositions" in propositional boolean formula
Suppose we had a TM $M$ which, given $F$ and some atomic $A$ occurring in $F$ it can efficiently (PTIME) decide whether all models of $F$ make $A$ true, i.e. whether $F \models A$.

Then we can efficiently decide the validity of any formula $G$ as follows.

Take $A$ to be any atom not in $G$. Craft $F := \lnot (G \land (\top \lor A))$, and use $M$ to decide if $F \models A$.

We then have 
$$
(F \models A) \iff (F \models \bot) \iff \models \lnot F \iff \models G \land (\top \lor A) \iff \models G
$$

The first $\iff$ is nontrival, but easy to prove by considering the models where $A$ holds, and the others where $\lnot A$ holds, and exploiting our choice of $A$. It is left as an exercise ;-)

So, we constructed a polynomial reduction of validity to this problem.
Since validity is coNP hard, we deduce this problem is also coNP hard.