## Theory of Complexity – Why Cook-Levin Thorem's Evidence May Mean NP-Hardness

I'm studying Cook-Levin's theorem, but I've encountered a problem. The Cook-Levin theorem shows that any NPTM can be encoded as a Boolean formula. About the given language $$A$$example $$w$$and NPTM $$M$$ Who's deciding $$A$$, I understand that when a Boolean formula $$φ$$ is true when $$M$$ accepted $$w$$ and wrong when $$M$$ rejects $$w$$, decision problem $$w∈A?$$ will be the same problem as $$φ = true?$$. But I am confused as to the fact that we can certify the existence of a reduction of any other problem from NP to SAT, with Cook-Levin's theorem. I can not make a real reduction from Karp to SAT without specifying the origin of the reduction, so I can not say that the SAT is in NP-Hard. Please, help me understand why conversion to Boolean formula can mean NP-Hardness of SAT …