I've made it to section 12 in Kleene's Mathematical Logic, which is about completeness. Surprisingly, I was able to understand how every valid formula is provable. However, one of the exercises he gives (12.4) is to show that every formula is provable after adding $A \supset B$ to the axioms of the propositional calculus. I understand that adding this axiom kind of breaks everything down, because it makes the deduction theorem a moot point. I've had a bit of trouble formalizing my thoughts though.
One of the theorems (theorem 10) says that if $\vdash A\supset B$, then $A \vdash B$. However, if $A \supset B$ is an axiom, then the hypothesis is always true and B is always provable. Is this all that's required to show that introducing implication as an axiom makes every formula provable? It seems right in my mind, but also "too easy," so I just wanted to get some verification.