Using home equity as down payment

[Jose A Perez]

Please read this plan of attack, which my loan agent wrote for my private lender and me. 

(Private Lender)- Purchases home at $100k

Renovates the home for $20k

ARV: 160k

I will buy the house from him @125k. 

He will gift me the $35k difference, which I will use as equity for a downpayment.

I'm confused about using the equity he gifts us as a downpayment to purchase this home. Would that mean we would refinance ourselves, taking $35k cash out of our home from HELOC or a Loan, then using the actual cash for the downpayment and have a separate loan for the full payment of the house? I see that our total loan would be 160k which could be considered refinance in a BRRRR, and I mean, the numbers work great with our plans. But how does someone use equity to buy a home???