Meaning of "Invece credo di sì"

Question

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?

Answer 1

In this context, it seems to be better translated as "in fact", "as a matter of fact"

Yes, here that's the meaning.

is "credo di sì" a more common answer than "credo che tu sappia" in this context?

Credo che tu sappia wouldn't work: something is missing. He might say Credo che tu lo sappia, but it would sound a tad too formal for an everyday exchange.

Answer 2

The adverb invece means al contrario, all’opposto. The

A translation into English would be

To the contrary, I think you do

Of course, other translations are possible, depending on the context (or dubbing needs). The site you reference is correct when translating Pensavo fosse a casa, invece non c'era. into I thought he would be at home, but he wasn't. This shouldn't make you believe that invece always means but or instead.