Die-hard and Deep down

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

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

Deep down adverb - In reality.

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