Homotopic paths and Deformation Theorem

Question

This is a general question about homotopic paths. Is it always true that if you have a simple, closed, positively-oriented curve $\gamma$, a point $a$ inside it anda function $f(z)$ which is holomorphic on and inside $\gamma$ then $\int_\gamma f(z)dz = \int_{\gamma_{(a,r)}}f(z)dz$, where $\gamma_{(a,r)}$ is a positively oriented circle centred at $\gamma$ with radius $r$ such that it is entirely in $\gamma$? I feel like this follows from the fact that a simple, closed curve divides the complex plane in $2$ sets by the Jordan's Curve Theorem and every $\gamma_{(a_1,r_1)}$, $\gamma_{(a_2,r_2)}$ are homotopic in the complex plane and hence in the interior. I can't seem to formulate it rigorously enough, I think in order to apply the Deformation Theorem. Any help is appreciated.