Doing as you propose will get you a $6,000 deduction on your tax return but the $10,000 regular distribution from the from the traditional IRA will add $10,000 to your taxable income. Generally, it would be better to have never made the $6,000 contribution, temporarily invested that $6,000 outside the IRA, then take only a $4,000 distribution from the IRA to which you would need to apply the first-time homebuyer's exception.
So no, if you are proposing just making a regular distribution of $10,000, it does not seem to be the sensible thing to do. In your situation it seems that it would be better to obtain a return of contribution of the $6,000 contribution and make a regular distribution of only $4,000, needing to apply the first-time homebuyer's exception to only the $4,000 plus any gains on the $6,000 distributed with the return of the $6,000. This preserves a large portion of your $10,000 lifetime limit on the use of the first-time homebuyer's exception to the 10% early-distribution penalty. By making a return of contribution (you must explicitly ask the IRA custodian to make a return of contribution) of the $6,000 you do not get the deduction but the $6,000 distributed as a return of contribution also does not get added to your taxable income.
Be sure to make the distributions shortly before the closing date of the purchase to give you the maximum window to treat the distribution as being for a first-home purchase.
