Intuitionistic logic contains the rule $\bot \rightarrow \phi$ for every $\phi$. In the formulations I have seen this is a separate axiom, and the logic without this axiom(?) is termed "minimal logic".
Is this rule required for practical proof problems in intuitionism? Is there a good example of a practical proof which goes through in intuitionism which doesn't go through without this axiom, or do all/most important practical results also go through in minimal logic? And can you please illustrate this with a practical example?
Context: We are faced with a decision theory problem in which it might be very useful to have a powerful reasoning logic which can nonetheless notice and filter consequences which are passing through a principle of explosion. So if the important proofs use a rule like $(A \vee B), \neg A \vdash B$ and we can replace that with $(A \vee B), \neg A \vdash \neg \neg B$ to distinguish the proofs going through the 'explosive' reasoning, that would also be useful.
ADDED clarification: I'm not looking for a generic propositional formula which can't be proven, I'm looking for a theorem in topology or computability theory or something which can be proven in intuitionism but not in minimal logic, along with a highlighting of which step requires explosion. Could be a very simple theorem but I'd still want it to be a useful statement in some concrete domain.