I heard the following dialogue in a TV series:
- Woman: Proverò a sentirlo, ma purtroppo non so dove sia.
- Gangster: Io invece credo di sì.
Context: The dialogue is about a priest who is missing. The woman tells the gangster she does not know where the priest is, but the gangster does not believe her and says the sentence above.
First, Word Reference (https://www.wordreference.com/iten/invece) says that "invece" means "but" / "instead", but it seems that this Italian word is also used when contradicting a previously mentioned fact in a separate sentence, unlike English "but". In this context, it seems to be better translated as "in fact", "as a matter of fact". Is that so?
Second, is "credo di sì" a more common answer than "credo che tu sappia" in this context?