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Can someone explain to me how the double negation of A is a premise for the negation of A? My understanding is this, if A is true then the double negation of A is A which is true, how is this a premise for negation of A?

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    Is this maybe part of a set up for a proof by contradiction? – 2017-01-31
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    Yes it is, I think that might be it – 2017-01-31
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    So does that answer your question or are you still confused about this? – 2017-01-31
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    will in line 2 where does double negation of A come from? i dont see it in the initial wff that im trying to prove if that makes any sense – 2017-01-31

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The $\neg \neg A$ is an additional assumption that starts a little subproof (hence it is indented). Since this additional assumption leads to a contradiction (false), we know that $\neg \neg A$ cannot be true, meaning that $\neg A$ is true. This proof technique (by proving that something is true by showing that its negation leads to a contradiction) is called a Proof by Contradiction or Indirect Proof (IP).

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    are you allowed to make any additional assumptions like this in a proof even if its not in the given statement? – 2017-01-31
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    @Code971 Yes. You can make *any* additional assumptions in a proof ... As long as you clearly indicate that (and here they do that by the indentation) so it is clear what the assumptions are behind each line in the proof ... Only the non-indented statements will be true merely on the basis of the original premises ... All others will depend on those additional assumptions. – 2017-01-31
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    man i am so dumb, i see what is going on now, they are just trying to derive false inorder to use IP. but thanks for your clarification tho. – 2017-01-31
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See:

$\sim(\sim A) = A,$ and

$A$ is not no $B = A$ is $B;$

All $A$ is not no $B$ = All $A$ is $B;$

Some $A$ is not no $B =$ Some $A$ is $B;$

None $A$ is not no $B =$ None $A$ is $B;$