Are the paradoxes of material or strict implication used anywhere to prove theorems in mathematics In the Stanford Encyclopedia of Philosophy entry "Relevance Logic", the following inference is listed as classically valid:

The moon is made of green cheese.  Therefore, it is raining in Ecuador now or it is not.

This inference can be tweaked slightly to make it more mathematical:

The moon is made of green cheese. Therefore, $\mathit{CH}$ or $\lnot\mathit{CH}$ (where $\mathit{CH}$ stands for the continuum hypothesis).

My questions are simply these:


*

*Are the so-called 'paradoxes' of material and strict implication ever actually used as legitimate steps in proofs of theorems in mathematics? (If so, give examples,e.g., proofs of theorems in $\mathit{PA}$, though proofs of this sort in any branch of mathematics would be nice as well.)  Also, if a proof of a theorem uses a 'paradox' of material or strict implication, can the proof be rewritten so as to avoid the use of the 'paradox'?  

*Consider the "variable sharing principle" from the above link, i.e., that "no formula of the form $A\rightarrow B$ can be proven if $A$ and $B$ do not have at least one propositional variable in common in common and that no inference can be shown to be valid if the premises and conclusion do not share at least one propositional variable."  Is this idea of relevance logic also part of standard mathematical practice?

*If it is helpful to distinguish 'mathematical inference' from 'logical inference', then:

What sort of relevance logic captures the notion of 'mathematical inference'?    

 A: In answer to 1:
Yes, there are theorems of PA that are not theorems of relevant PA.  Harvey Friedman's example is "for every odd $n$ there is an integer which is not a quadratic residue mod $n$", or $\forall x\ \exists y\ \forall z\ \exists a\ \exists b\ (2x+1)a + (y - z^2)b = 1$.  
The proof relies on this being a statement with no negation which fails over the complex numbers, and uses Tarski's result on the undefinability of countably infinite sets in that field.  See Meyer and Friedman, Whither Relevant Arithmetic, Journal of Symbolic Logic 1992.
And yes, proofs can be rewritten to be relevantly valid.  The same paper also proves that if T is a theorem of PA then "T or not 0=0" is a relevant theorem of PA.  
In answer to 2:
Apparently some ordinary mathematics does not abide by the rules of relevant logic.  
Commenting on 3 without answering it:
I would say that the statement 0=0 is relevant to all arithmetical statements (while the rainfall today in Ecuador might not be relevant to all astronomical statements).  So I do not see the value of using relevant logic in the arithmetical context. 
