So if I want to prove. A=>B for all x.
Does the following work?
Suppose for contradiction, B is not true for all x, that is, there exists at least one x such that B is not true. In particular, assume that B is true for x=c and B isn't true for all other x. If I arrive at a contradiction, then A must imply B.
So does it work if I pick a single value of x such that B is true and let B not be true for all other values? This is a little confusing because the negation simply specifies the case for at least one x such that B isn't true. There could be more than one x such that B isn't true.