How to prove two statements are equivalent? I got into a bit of a disagreement. How could I prove or disprove that theses two statements are equivalent: 

Atheism is the disbelief in the existence of a god or gods 

and

Atheism is the belief that god or gods don't exist.

Those are dictionary definitions for Atheism.
I think I may be over complicating things. All I'm asking is how to prove two statements are equivalent using logic.
 A: The two statements are not equivalent. (Study the difference between proving a mathematical statement and disproving it.) Buddhism, for example, (to my knowledge) does not purport the existence of a deity, but it is not atheism.
A: The issue I see here is that, does disbelieving in a concept mean that you purport that the concept is false?  Here the statements don't seem equivalent since someone not believing in the existence of god(s) does not necessarily mean they insist on god(s) not existing.  To clarify, the latter statement implies the former but not the other way around.  If atheists believe god(s) don't exist, then certainly they are in the disbelief god(s) existing.  However, if an atheist is in the disbelief of god(s) existing, then that does not necessarily mean they believe the opposite.  
A simpler example of this same logical inequivalence: If I'm making lasagna I need to boil pasta, but if I'm boiling pasta I'm not necessarily making lasagna.  
A: You can change these into two propositions. The statements themselves don't appear to me as propositions. But, what if you changed them into propositions: If Atheism is true, then god doesn't exist. (Atheism isn't a claim about belief, it is a claim that God does not exist). You're are asking to tell what this is logically equivalent to, if you agree with how I rephrased it into a proposition. 
Let P be the proposition that Atheism is true, and Q be the conclusion that god doesn't exist. 
But, P--->Q
is the same as it's contrapositive, or, not Q ----> not P
which is, if god exists, then atheism is false. I hope this helps. 
A: Let's keep religion out of it. You are asking about the distinction between statements of the following forms:


*

*I don't believe my next-door neighbours have a cat.

*I believe my next-door neighbours don't have a cat.
Whether these two statements are equivalent depends on what kind of doxastic logic you are working with.
A: The "disbilief in the existence" approach has the subtle implication of not believing in spite of an object being real, whereas the "belief that $x$ does not exists" is not compromised with existence of any kind. 
The argument I'm trying to give is that: 1) each approach has its own emphasis (were it in an epistemic attitude of denial or resistence, or a doxastic state of affairs in an agent's web of beliefs); and 2) as others have pointed out, it is not clear those kinds of propositions are equivalent, since there are many epistemic or doxastic logics that solve the problem differently on the basis of their proper semantics of modal operators as "believe in" and "not believe in", etc.
A: Statement 1:
Atheism is the disbelief in the existence of a god or gods
=>Athiesm is the (non-belief) [in the existence of a god or gods]
=>Athiesm is [ (non-belief) (in the existence) (of a god or gods)]
=>Athiesm is the belief in[(non) existence of a god or gods]
"opposite of an opposing statement is the same statement!"
=>Athiesm is the belief in (non-existence) of a god or gods
=>Athiesm is the belief of a god or gods' (non-existence)
=>Atheism is the belief that god or gods don't exist.
=>Statement 2
Hence proved.
