I am assuming that we are talking about taking the law of excluded middle out from classical logic, where all the other things independent from that law still hold, and I think implication being material is independent.
In propositional logic, material implication is a valid rule of replacement which is an instance of the connective of the same name. It is the rule that states that "P implies Q" is logically equivalent to "not-P or Q".
Nice flame, eot.
Arguably, the common sense definition of 'A -> B' is simply modus ponens. That is 'Given A and A->B, it follows that B'. It's then no longer obvious that A -> B is equivalent to ~A v B.
Indeed, implication in IL is different from material implication even though IL has modus ponens.