No, you can't. Two mathematicians using different rules of inference (say, those of classical vs. intuitionistic logic) will arrive at different theorems, even if they begin from the same axioms.
> "Assuming a=0 implies a=0" is absolutely true.
Assuming that “a = 0” and implication are both expressible in our formal system, it would indeed be very weird (though not a priori impossible) if “a = 0” didn't imply “a = 0”. But I have no problem imagining a formal system where “a = 0” isn't expressible (e.g., because equality is inexpressible), or implication isn't expressible (e.g., geometric logic), or implication has strange properties to someone only familiar with the classical and/or intuitionistic interpretation of “implication” (e.g., linear logic).