Simple Question on Contrapositive Proof If I want to prove something along the lines of:
If there exists a j which satisfies the conditions:
1)... 2)... 3)... then Something awesome happens.
I proved the forward direction of this (its an iff proof)
I want to prove the reverse direction
How do I do it using contrapositive?
I understand that a contrapositive of 
X > Y is ~Y>~X.
So, in my case, I should have:
something awesome > existence equivalent to ~existence > ~ something awesome.
But, what is ~existence? How do i connect the 3 conditions with ~existence?
 A: For proving $iff$, letting $P_j$ denot "$p$ is prime," i.e., to prove: 
$$\exists j(Pj \land j\neq 2 \land j\in[1, 100]) \iff (\text{blah happens}),$$ 
and given that you've established/proved the forward direction ($\implies$), if you want to prove the reverse implication
$$(\text{blah happens}) \implies \exists j(Pj \land j\neq 2 \land j\in[1, 100])$$
by proving its contrapositive, 
then you would need to establish that $$\lnot \exists j(Pj \land j\neq2 \land j\in[1,100]) \implies \lnot \text{(blah happens)},\tag{1}$$ 
or equivalently 
$$\forall j (\lnot Pj \lor j = 2 \lor j\notin[1,100]) \implies \lnot \text{(blah happens)}.\tag{2}$$ 

EDIT  as requested by the OP:
To understand why $(1)$ and $(2)$ are equivalent, with the understanding that each represents the contrapositive of $$(\text{blah happens}) \implies \exists j(Pj \land j\neq 2 \land j\in[1, 100]) $$
We start from $(1)$:
$$\lnot \exists j(Pj \land j\neq2 \land j\in[1,100]) \implies \lnot \text{(blah happens)}.\tag{1}$$ 
We switch to the universal quantifier while moving the negation inward gives us the equivalent statement:
$$\forall j[\lnot(Pj \land j\neq2 \land j\in[1,100])] \implies \lnot \text{(blah happens)}.\tag{1.a}$$ 
Then by DeMorgan's law $(1.a)$ is equivalent to $(1.b)$:
$$\forall j[(\lnot Pj \lor (\lnot (j\neq2) \lor (\lnot(j\in[1,100]))] \implies \lnot \text{(blah happens)},\tag{1.b}$$
which you can see is equivalent to $(2)$:
$$\forall j[(\lnot Pj \lor (j = 2) \lor (j\notin[1,100])] \implies \lnot \text{(blah happens)}\tag{2}$$ 
