Effective Procedures and P vs NP Problem
No. $\mathrm{P}\neq\mathrm{NP}$ means that there is no efficient (i.e., deterministic polynomial time) solution to $\mathrm{NP}$-hard problems, not that there is no effective (i.e., computable) solution. The fact that the solution is hard to find doesn't mean that it's impossible to find. Indeed, essentially the whole reason that $\mathrm{NP}$ is an interesting class of problems is that we know how to check proposed solutions to $\mathrm{NP}$ efficiently. Regardless of how hard it is to find satisfying assignments to Boolean formulas, if I give you an assignment and tell you that it satisfies the formula, you can easily check that I'm telling the truth.