Elimination question on alkyl halide

[Zayan]

Homework Statement

Find the major product

Relevant Equations

E1/E2/Sn1/Sn2

I firstly made the newman projection, aci/base on OH resulting in OH2+. Then NGP by neighbouring bromine and then the other bromine back attack. Then I simply did Finkelstein reaction and hence anti eliminating erythro iodine from vicinal positions making Trans 2 butene. The solution manual however has done something different. Normal external SN2 followed by Iodine removal(see the image). But the have done the anti elimination on threo iodine. It should give cis alkene, but they have given answer as trans alkene. So either I am somehow wrong, or the solution manual has done wrong mechanism. Please help.

The above is the question.

This is the solution provided.