NOT satisfiable 3SAT instance certificate
A CNF which is not satisfiable is usually called unsatisfiable. A CNF which is unsatisfiable but becomes satisfiable if we drop any clause is minimally unsatisfiable.
Papadimitrious and Wolfe constructed, in their paper The complexity of facets resolved (Lemma 1), a polynomial time reduction $f$ from CNFs to CNFs such that:

If $\varphi$ is satisfiable then $f(\varphi)$ is satisfiable.
If $\varphi$ is unsatisfiable then $f(\varphi)$ is minimally unsatisfiable.

It is likely that this can be extended to a reduction from 3CNFs to 3CNFs.
This reduction shows that the minimally unsatisfiable case is the &quot;hardest&quot;. In particular, if there is an efficient way to check that a minimally unsatisfiable CNF is unsatisfiable, then P=NP.