Let $I$ be the formula which states that there exists strongly inaccessible cardinals.
My question is regarding the proof of $ZFC\nvdash I$ appearing in Jech (part of theorem 12.12). He starts by proving (in $ZFC$) that if $\kappa$ is strongly inaccessible then $V_\kappa\models ZFC$. Why is it not done here? if $ZFC\vdash I$ then $ZFC$ proves its own consistency ($V_\kappa$ is a model) which contradicts the second incompleteness theorem.
Instead, he proceeds to claim that $V_\kappa\models \neg I$ (which requires some effort i think), and then says that if $ZFC\vdash I$ then any model for $ZFC$ is also a model for $I$, which contradicts $V_\kappa\models ZFC,\neg I$. Is this necessary? or can i stop after $V_\kappa \models ZFC$?
Your argument is correct but the point is that you do not have to appeal to Godel's 2nd incompleteness theorem to prove this and Jech does that.