 Solution

Use the Hilbert-Bernays-Löb derivability conditions for an arithmetical provability predicate. Abbreviate $P_S(┌A┐)$ by $PA$. The conditions needed below areThese are the usual theoremhood, distribution and positive-introspection conditions intended by a standard arithmetical provability predicate; arbitrary predicate symbols merely agreeing with provability externally need not satisfy them.

Since $S⊢\neg Y$, it proves $Y\to G$. The first two conditions give $S⊢PY\to PG$. For the converse, the fixed-point equivalence gives $S⊢G\to (PG\to Y)$. Applying theoremhood and distribution givesPositive introspection also gives $PG\to PPG$, and distribution combines these to yield $S⊢PG\to PY$. Thus $S⊢PG\leftrightarrow PY$. Combining with $G\leftrightarrow \neg PG$ provesIf $S$ proved $\neg PY$, it would prove $G$. The theoremhood condition would then give $PG$, while the fixed-point equivalence gives $\neg PG$. Therefore first-order consistency implies