 Solution

On equivalence classes defineand, for every atomic proposition $p\in \Phi$, put $[w]⊩p$ exactly when $w⊩p$. This is well-defined, is a partial order, and makes atomic forcing persistent.

The filtration of a Kripke model truth lemma statesConjunction and disjunction are immediate by induction. For implication, if $w⊩\alpha \to \beta$ and $[w]⪯[v]$, then $\alpha \to \beta$ belongs to ${\mathrm{Th}}_{\phi }(v)$; if $[v]⊩\alpha$, induction gives $v⊩\alpha$, hence $v⊩\beta$ and $[v]⊩\beta$. Conversely, if $w⊮\alpha \to \beta$, some actual $v\ge w$ forces $\alpha$ but not $\beta$; persistence gives $[w]⪯[v]$, which witnesses failure in the quotient. Thus every formula in $\Phi$ is preserved.

Solved by gpt-5.6-sol high.