If 1 day of dry fasting is equivalent to 3 days of water-only fasting, induction into ketosis is not one of the ways in which this is true since carb stores in the body are not being depleted any faster. Thus in answer to your question, going into ketosis in a dry fast should take about 2 or 3 days, just as in a water-only fast.
If dry fasting is 3x faster (or approximately so) in anything, it is in the cleansing and healing accomplished through it, by way of the partially dehydrated cells which burn up toxins like a furnace.