Soundness of Ω-logic: $T \vdash_\Omega \varphi \Rightarrow T \models_\Omega \varphi$

Soundness of Ω-logic (Woodin; primer, Theorem 3.3). Assume there is a proper class of strongly inaccessible cardinals. Then for every set of sentences $T$ and every sentence $\varphi$,

That is, if a universally Baire set $A$ has the property that $M_\alpha \models T$ implies $M_\alpha \models \varphi$ for every strongly $A$-closed countable transitive model $M$ of ZFC and every ordinal $\alpha \in M$, then $V^B_\alpha \models T$ implies $V^B_\alpha \models \varphi$ for every complete Boolean algebra $B$ and every ordinal $\alpha$.

This is the "if" direction of the Ω-conjecture.