Deep down and Die-hard

Deep down adverb - In reality.

Die-hard and deep down are semantically related. in deep-rooted topic. In some cases you can use "Die-hard" instead an adjective phrase "Deep down".

Die-hard adjective - Tending to favor established ideas, conditions, or institutions.

Deep down and die-hard are semantically related. Sometimes you can replace phrase "Deep down" with "Die-hard", this phrases are similar.