Most often when a certain axiom or proposition implies another proposition, this implication is not trivial, or "immediate". That is, you need a proof consisting of some number of steps to reach the second proposition.
So in that case we can say, proposition A implies but does not trivially imply proposition B.
My (philosophical) question is: what does it mean for a proposition to trivially imply another, or in other words, that it follows "immediately"? Is there a way to objectively determine whether an implication is trivial? Does a trivial implication only rely on an appeal to an intuition about "primitive notions"?
Perhaps we should exclude from this analysis cases where a theorem is implicitly used but simply not explicitly stated for the sake of brevity.