Op Amp magnitude of gain proof
Bringing the resistances out


Starting from your final equation,
$$\frac{V_o}{V_i} = \frac{1}{\frac{R_1}{R_f}-j\frac{f_c}{f}}\tag{1}$$
 where \$f_c =\dfrac{1}{2\pi R_f C}\$. 

Let \$f_1 = \dfrac{1}{2\pi R_1 C}\$ then,
$$f_c = \frac{R_1}{R_f}\times f_1$$ Applying this in \$(1)\$,
$$\frac{V_o}{V_i} = \frac{1}{\frac{R_1}{R_f}-j\frac{f_1}{f}\times \frac{R_1}{R_f}}$$
$$\left|\frac{V_o}{V_i}\right| = \left|\frac{R_f}{R_1}\times\frac{1}{1-j\frac{f_1}{f}}\right|$$
$$= \frac{R_f}{R_1}\frac{1}{\sqrt{1+\frac{f_1^2}{f^2}}}$$




  Alternate method


You should have started this way. Especially when you had the final answer with you. :)

$$ \frac {V_o}{V_i} = \frac {R_f}{R_1 - \frac {j}{\omega C}}$$
$$= \frac{R_f}{R_1}\frac{1}{1-j\frac{1}{\omega R_1 C}}$$
$$= \frac{R_f}{R_1}\frac{1}{1-j\frac{f_1}{f}}$$

Taking absolute value results in the required result.