I'm struggling with a proof in the text for my logic course, and I'm wondering if someone could offer a hint or some help. The question is basically as follows. Show that if the decision problem for satisfiability is solvable iff the decision problem for implication is solvable. Then, show that the decision problem of implication is solvable iff the decision problem of validity is solvable.
My attempt so far is as follows. Suppose we can decide whether or not a given sentence is satisfiable. If not, implication problem for the sentence A is true. If it is satisfiable, we must ensure that all interpretations in which A is satisfiable, B is also true.
I'm assuming the second part proceeds similarly, but I'm not exactly sure, and would appreciate any guidance.