OK, here is the recommended spec list for Adobe Premiere CC 2017 - https://helpx.adobe.com/premiere-pro/system-requirements.html I am not a user of this software, but I find it bizarre that most current GPU's don't appear to be supported. These are all a few generations old. An older version, CC 2014 did apparently support the HD7870, but it's not listed for CC 2017. While reading around, I found that on CS6, adding GPU support may be achieved in some cases by simply editing a text file to add the name of your GPU to it... I'm not sure if this works on CC. You perhaps need to clarify with Adobe themselves or a Premiere user group on what the situation is.
Anyway, as a starting point, I think something like this is reasonable for a fast machine that could do what you wanted. I'm not an expert on what you'd need in a high level video editing machine, but I should think this would give good performance for anything you could throw at it. You'd need an operating system as well, unless you can transfer your old one (non-OEM licenses only).
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Generated by PCPartPicker 2017-05-18 11:26 EDT-0400
Ryzen 7 1700 - Best price performance ratio for this type of work, IMO
16 Gb RAM - Enough, and this particular module is designed to work with Ryzen CPU's (they are known to be fussy about memory)
Motherboard - It's no frills but quite well featured, by a reputable manufacturer. Supports USB 3.1 for adding a high end external disk enclosure if required.
Storage - Fast PCIe SSD for snappy performance. Generous 3 Tb disk for storing large video projects.
GPU - simply, it's one of the few that are listed as compatible with Premiere that are available to buy. You may save money by re-using your HD7870 if it can do what you need. Otherwise you may get better performance in a more modern design (e.g. GTX 1060, RX-580) if they support Premiere.
PSU - more than enough power for the job. Pretty good quality and warranty.
Case - Personal choice. Corsair are pretty good though.
Edited by jonuk76, 18 May 2017 - 10:30 AM.