Proving theorems using predicate calculus laws

Question

[ ¬P => (P => Q) ]

[ (P => Q) => (P => (Q \/ R)) ]

How would I prove these 2 theorems?

Answer 1

For the first one, let's start with the fact that p --> q is true iff q v ~p (this is the rule of Material Implication, but you can also just think of the truth tables—they're the same!). so if we have ~p, we can get ~p v q by weakening/disjunction intro. And ~p v q is equivalent to p --> q by material implication again. et voila!

I'll leave the second to you, but it's similar—it even depends on weakening/disjunctive intro like #1.

Hope that helps!