 As it is evident that \(g(x,y)\) is partial computable, \(g(x,y) \simeq \phi_e(x,y)\) for some \(e\). It thus follows by the  s-m-n Theorem (3.1)  that \(\phi_e(x,y) \simeq \phi_{s^2_1(e,x)}(y)\). Let \(b(x) = s^2_1(e,x)\) and note that we then have \(\phi_{b(x)}(y)\) is the same function as \(\phi_{\phi_x(x)}(y)\) provided that \(\phi_x(x)\) is defined. Note that \(b(x)\) is a total computable function and is defined independently of the given function \(f(x)\).