How to prove that \$I_3=Vi/R1\$ in a darlingtonpair circuit with opamp when \$\beta_2\to\infty\$?
You don't need to prove it if you accept that an op-amp (even quite modestly priced op-amps) that use negative feedback, seek to ensure that the two input voltages (+Vin and -Vin) are identical.
In effect, what ever voltage you apply to +Vin (Vi in your example) results in the same Vi appearing on -Vin. This inevitably means that: -
$$I_{R_3} = \dfrac{V_i}{R_1}$$