In everyday talk we say things like: we must use the fundamental theorem of calculus to calculate this integral; we must use the results of analysis to proof the fundamental theorem of algebra; we must have a function that maps well formed formulas to numbers and that is bijective (Godel beta function); we must have a ring with the fundamental theorem of algebra to be able to proof this result; etc...
Sometimes it seems that, when we say this, we are saying that the theorem in question must be made an axiom, then we will try to investigate what results hold with it, or without it, like in the case of the fundamental theorem of algebra. Sometimes it seems that we are saying that to proof some theorem we need to use a central theorem that mobilizes a particular set of axioms, this seems to be the case with the fundamental theorem of calculus, or the case of a theorem that say that some function or property exist and we must use it.
In this last case, which is the case I am interested, it seems that the construction we are trying to make needs another construction, that in turn needs a set of axioms. But it's not the case that, if we simply fix it as an axiom, we could see what thing depends or not on it, for fixing it as axiom would turn the axiom system redundant (meaning: one or more axioms could be eliminated).
So, my question is: what means to say that a theorem depends on another theorem? Is there some logical analysis of what means to say this?