Open Mapping Theorem: let $X$ and $Y$ be Banach spaces and let $T : X \to Y$ be a linear surjective continuous map. Then $T$ is opened.
Definition: let $X$ and $Y$ be two normed spaces and let $T : X \to Y$ be a mapping. We say that $T$ is an isomorphism if it is continuous, opened and bijective.
Banach Isomorphism Theorem: let $X$ and $Y$ be Banach spaces and let $T : X \to Y$ be a linear bijective continuous map. Then $T$ is isomorphism.
Is it possible to show the Open Mapping theorem using Banach Isomorphism theorem? I think not because the hypothesis from the second is more specific (where do we get the injectivity from?); in fact, the second one is a trivial corollary from the first one: if we take the hypothesis from the Banach Isomorphism theorem, using the Open Mapping theorem, we have that $T$ is is opened, it means if $T$ is furthermore bijective that $T^{- 1} : Y \to X$ is continuous.