Is a logical disjunction statement reversible? This might be a stupid question, but I'm just learning proofs so I'm unsure.
if I have $e \vee f$, can I change it to $f \vee e$ without repercussion?
 A: Yes you can, because the statement $A\lor B$ is true if and only if the statement $B\lor A$ is true.
In other words, if you have $A\lor B$, you can prove $B\lor A$, and vice versa.
A: Either this or that.  Is that the same as either that or this?  If either this or that is true than both statements are true.  If neither this nor that is true than both statements are false, so they are equivalent.
Let's look at truth values.  (Sorry I don't know how to use LaTex to make tables)
this TRUE; that TRUE; $\rightarrow$ this $\vee$ that TRUE; that $\vee$ this TRUE
this TRUE; that FALSE; $\rightarrow$ this $\vee$ that TRUE; that $\vee$ this TRUE 
this FALSE; that TRUE; $\rightarrow$ this $\vee$ that TRUE; that $\vee$ this TRUE 
this FALSE; that FALSE; $\rightarrow$ this $\vee$ that FALSE; that $\vee$ this FALSE 
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Or another way to think of it.  The rules are axiomatic.  But they are axiomatic to represent real meaning.  "and" and "or" don't have any meaning change when the order is reversed.  So they are commutative (do we use that term in logic?).  
But as one commentator pointed out, as these are axiomatic, we can imagine constructed systems with different axioms where this isn't the case.  But that's a different subject. 
A: One way to quickly check such statements is to place them in a truth table generator and see if a tautology results. 
Consider this truth table:

For all valuations of $e$ and $f$, the biconditional column shows "T". That makes this a tautology and confirms that one can replace $e \lor f$ with $f \lor e$ without concern. However, if one is using specific inference rules, one may or may not be able to go from one to the other immediately.

Michael Rieppel. Truth Table Generator. https://mrieppel.net/prog/truthtable.html
