Priority claims are governed by the Paris Convention art 4.
Arts 4(C)(1)–(2) requires that priority can be claimed for twelve months from the date of filing the first application.
Art 4(C)(4) provides that a subsequent application can only be used if the first filing "has been withdrawn, abandoned, or refused, without having been laid open to public inspection and without leaving any rights outstanding, and if it has not yet served as a basis for claiming a right of priority".
Thus given your example timeline:
1 Jan 2010: US provisional
1 Jul 2010: US non-provisional (claiming the priority of US provisional)
1 Sep 2010: WO application (claiming the priority of US non-provisional)
The WO application could validly claim priority from the US provisional (up until 1 Jan 2011, 12 months from the first filing). It could also claim priority from the US non-provisional (again, up until 1 Jan 2011, 12 months from the first filing). Since the WO was filed within this period, it has a valid priority claim.
The curious case of WO 1999/052812
You mentioned this case in your comment. The family timeframe of this appears to be:
22 April 1997: Provisional US 60/044588 was filed.
16 April 1998: Non-provisional US 09/061823 was filed (which is seemingly identical to the provisional).
12 April 1999: WO 1999/052812 is filed, claiming priority to US 09/061823.
This appears to be a clear case of an invalid priority claim.
The provisional could be used for priority only until 22 April 1998 (12 months from filing). Since the non-provisional has the same subject matter, and the provisional wasn't first abandoned, the non-provisional doesn't reset the clock. The WO filing therefore could only claim priority (to the provisional or the non-provisional) until 22 April 1998. Since it was filed in 1999, it is well beyond the priority period, and thus has an invalid priority claim.
This doesn't seem to have been noted by the USPTO (acting as ISA on the WO) or the EPO (in the subsequent EP regional phase). I can only speculate why, but I presume it's simply because nobody noticed.