By Corollary 5.8, the Banach algebra norm $\vvvert\mathord{\cdot}\vvvert_{\phi_{X|Y}}$ is an affinoid algebra norm. By Proposition 2.58, there exists $C(\phi,Y)>0$ such that

$$\vvvert\mathord{\cdot}\vvvert_{\phi_{X|Y}}\leq C(\phi,Y,X)\cdot\vvvert\mathord{\cdot}\vvvert_{\phi_{X|Y};\mathrm{sp}}.$$

Since $\vvvert\mathord{\cdot}\vvvert_{\phi_{X|Y};\mathrm{sp}}=\vvvert\mathord{\cdot}\vvvert_{\phi|_{Y}}$ by Corollary 3.27, one gets the bounds. ∎