How do we go from $P \lor Q = \lnot (P \downarrow Q)$

$P\lor Q = (P\downarrow Q) \downarrow (P\downarrow Q)$ to this
I understand they are equivalent, but I don't get how it went from this $\lnot (P \downarrow Q)$ to $(P\downarrow Q) \downarrow (P\downarrow Q)$. To be more specific let's say I wanted to see how you added 450 and 900 together, did you put 450 first then added 900, did you use mental math, did you regroup, etc. I am wanting to see the mathematical steps basically of how it was done. This is also straight out of Daniel J Velleman's "How To Prove It: A Structured Approach"