my thoughts on this are as follows.
i would put the cash-out into a separate a/c just to keep things clean should the IRS ever question what you did with it. This is what is referred to as the tracing rules. for the period the cash-out portion is not used for another property or investment purpose (such as buying stocks but this portion would then be investment interest subject to investment interest deduction limitations - schedule A), the interest would be nondeductible personal interest. so when it comes time to buy the new property putting in personal cash on top of the cash-out should be easy to demonstrate should the IRS inquire
