We apply classical proof complexity ideas to transfer lengths-of-proofs lower bounds for a propositional proof system P into examples of hard unsatisfiable formulas for a class Alg(P) of SAT algorithms determined by P. The class Alg(P) contains those algorithms M for which P proves in polynomial size tautologies expressing the soundness of M. For example, the class Alg(F"d) determined by a depth d Frege system contains the commonly considered enhancements of DPLL (even for small d). Exponential lower bounds are known for all F"d. Such results can be interpreted as a form of consistency of P NP. Further we show how the soundness statements can be used to find hard satisfiable instances, if they exist.