 Note that \(f(x,y)\) is partial computable since it is defined by cases in terms of \(\phi_a(x)\) based on the value of \(\phi_x(x)\). There is thus an index \(c\) such that \(f(x,y) \simeq \phi_c(x,y)\). By applying the  s-m-n Theorem (3.1),  we thus have that \(\phi_c(x,y) \simeq \phi_{s^2_1(c,x)}(y)\). But note that we now have the following sequences of implications: