I am trying to prove that $G=\langle \alpha,\beta,\gamma \mid \alpha\beta\alpha^{-1}\beta^{-1}\gamma \rangle$ is isomorphic to $H=\langle \delta,\varepsilon \mid \hspace{0.5cm}\rangle$. Let N be the smallest normal subgroup including $\alpha\beta\alpha^{-1}\beta^{-1}\gamma$
In order to prove it, I define a surjective map from $\langle \alpha,\beta,\gamma\rangle$ to $\langle \delta,\varepsilon\rangle$ and I try to apply the First Isomorphism Theorem.
$\varphi:\langle \alpha,\beta,\gamma\rangle \longrightarrow \langle \delta,\varepsilon\rangle$ with $\varphi(\alpha)=\delta,\varphi(\beta)=\varepsilon,\varphi(\gamma)=\varepsilon\delta\varepsilon^{-1}\delta^{-1}$. $\varphi$ is clearly surjective and $\varphi(\alpha\beta\alpha^{-1}\beta^{-1}\gamma)=\delta\varepsilon\delta^{-1}\varepsilon^{-1}\varepsilon\delta\varepsilon^{-1}\delta^{-1}=e$. So $N\subset\text{Ker}(\varphi)$
How do I prove $\text{Ker}(\varphi)\subset N$?