(⇐) The hypothesis for this half of the proof is that the system $\linearsystem{A}{\vect{b}}$ has a unique solution for every choice of the constant vector $\vect{b}$. We will make a very specific choice for $\vect{b}$: $\vect{b}=\zerovector$. Then we know that the system $\linearsystem{A}{\zerovector}$ has a unique solution. But this is precisely the definition of what it means for $A$ to be nonsingular (Definition NM). That almost seems too easy! Notice that we have not used the full power of our hypothesis, but there is nothing that says we must use a hypothesis to its fullest.
(⇒) We assume that $A$ is nonsingular of size $n\times n$, so we know there is a sequence of row operations that will convert $A$ into the identity matrix $I_n$ (Theorem NMRRI). Form the augmented matrix $A^\prime=\augmented{A}{\vect{b}}$ and apply this same sequence of row operations to $A^\prime$. The result will be the matrix $B^\prime=\augmented{I_n}{\vect{c}}$, which is in reduced row-echelon form with $r=n$. Then the augmented matrix $B^\prime$ represents the (extremely simple) system of equations $x_i=\vectorentry{\vect{c}}{i}$, $1\leq i\leq n$. The vector $\vect{c}$ is clearly a solution, so the system is consistent (Definition CS). With a consistent system, we use Theorem FVCS to count free variables. We find that there are $n-r=n-n=0$ free variables, and so we therefore know that the solution is unique. (This half of the proof was suggested by Asa Scherer.)