Provability and satisfiability are distinct, though. Satisfiability is concerned with a single formula, not a relationship between two formulae. For instance, I know that \\(\bot \vdash A \land \neg A\\), but \\(A \land \neg A\\) is not satisfiable.

Worse, satisfiability is weaker than validity, so it wouldn't even be fair to characterize satisfiability by \\(\top \vdash P\\).