1031 (or similar) selling rental for down payment on primary?

[Joe Edwards-Hoff]

Is there any tax strategy to not pay capital gains tax if I wanted to sell a rental house and use the equity to help with the down payment in a primary residence?

We're looking at a home that it more expensive than we'd typically think about, but if we sold our primary home and one of our rentals, we could pay it down substantially.  

(Also, I assume the answer is no, but is there any similar strategy to sell a property and use those proceeds to pay down the mortgage on another investment property?)

In my limited experience I don't believe either is possible,  but I'd hate to not ask and miss out.