NP⊕P⊆BPP⊕P implies PH⊕P⊆BPP⊕P (relativize 2 to ⊕P)
PH⊕P⊆BPP⊕P (use 5 and 6)
PH⊆BPP⊕P (immediate corollary of 7)
We often call results like Zachos (2 above) a "pigs
can fly" theorem because we don't believe the assumption in this
case that NP is in BPP. This proof shows that relativization can give
pigs wings and lead to some interesting containments.