I don't think these people noticed you also want to stream as well as game. so you should go with a ryzen based system you'll benifit from the extra cores and threads.
PCPartPicker part list:
https://pcpartpicker.com/list/c3RhP3
Price breakdown by merchant: https://pcpartpicker.com/list/c3RhP3/by_merchant/
CPU: AMD - Ryzen 7 1700 3.0GHz 8-Core Processor ($287.97 @ Amazon)
Motherboard: MSI - B350 TOMAHAWK PLUS ATX AM4 Motherboard ($66.98 @ Newegg)
Memory: Team - Dark 16GB (2 x 8GB) DDR4-3000 Memory ($165.75 @ Newegg)
Storage: SanDisk - Ultra 3D 500GB 2.5" Solid State Drive ($134.99 @ B&H)
Storage: Seagate - Barracuda 2TB 3.5" 7200RPM Internal Hard Drive ($54.99 @ Amazon)
Case: Phanteks - ECLIPSE P400S TEMPERED GLASS ATX Mid Tower Case ($59.99 @ Newegg)
Power Supply: SeaSonic - FOCUS Plus Gold 650W 80+ Gold Certified Fully-Modular ATX Power Supply ($83.99 @ SuperBiiz)
Total: $854.66
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2018-02-13 02:37 EST-0500
You'll have to wait for gpu prices to come down before you decide on a card