Let $K$ be a number field. $E/K$ be an elliptic curve over $K$. Let $\phi : E \to E'$ be an isogeny of degree 2, that is, $\phi・\hat{\phi}=[2]$. Let $\hat{\phi}$ be dual isogeny of $\phi$.
There is a map $f: E'(K)/\phi(E(K))\to E(K)/2E(K)$.
I want to prove the kernel of $f$ is exactly $E'(K)/\phi(E(K)[2])$. How can I prove this formally ? Thank you in advance.