Help me understand the logic behind x - y in binary by boolean?

Question

x and y are 4 bit signed numbers (2s complement..)

x - y can be obtained by:

!(!x + y)

I know that in 2s complement

-y = (!y + 1)

So I can pretty much understand how this works:

x - y = x + (!y + 1)

But, what sort of magic is going on in here, how does this work:

!(!x + y)

I mean, when I do the math with pen and paper all is fine, but I do not know why it is working..

Answer 1

If $z$ is an $n$-bit number then $!z=2^n-1-z$. So $$ !(!x+y)=2^n-1-(2^n-1-x+y) = x-y. $$