"State of the art" algorithms deciding entailment of propositional formulas?
Checking whether $P \Rightarrow Q$ is the same as checking that $P \land \lnot Q$ is unsatisfiable. Using the Tseytin transformation, you can convert $P \land \lnot Q$ into a form suitable for a SAT solver.
Conversely, satisfiability is a special case of your problem, since $P \Rightarrow \bot$ iff $P$ is unsatisfiable.