While this relationship between a valid argument and its "corresponding" conditional holds in standard first-order logic (and in a number of other systems) it does not hold generally for arbitrary systems of deductive logic.  That it does hold in fact is an interesting result for a deductive system, and it is called "the deduction theorem" (see Wikipedia and any number of other places, including virtually any text book on advanced logic or formal semantics).

There are a number of perfectly respectable systems in which the deduction theorem fails -- that is, in which an argument from the premise P to the conclusion S is valid but in which the conditional 'P->S' is not a theorem. Indeed, systems which attempt to capture a sense of 'if ... then' not represented by the material conditional (e.g., subjunctive conditionals) will often be ones in which the deduction theorem does not hold.  Some philosophical reflection may then be needed to justify such a system -- or alternatively, to justify a proposed requirement that the deduction theorem hold in order for a system to genuinely capture a sense of 'if ... then'.

One thing that this failure of the universal correspondence between a valid argument and a "corresponding" conditional illustrates is that the question of the difference between an argument and a conditional is (semantically or pragmatically) a bit deeper than it may first appear.