Starting with Xcode 7.3, in addition to copying in the SDK, you must also edit a certain Info.plist file, as described here for macOS in the post by agx. It looks like there is a similar file for iOS, at
Xcode.app/Contents/Developer/Platforms/iPhoneOS.platform/Info.plist
Try changing the value of MinimumSDKVersion in there from 10.0 to whatever SDK version you want.
I've not tried this with iOS, but, using this workaround, I was just able to build a macOS target with macOS 10.6 using Xcode 8.0 (8A218a) (the "GM").
UPDATE
The hack described above stopped working for me in Xcode 9. If you want to use Xcode 9+, see my more recent answer dated Nov 22, 2017.
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