Chemistry Elimination question on alkyl halide

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The discussion centers on a discrepancy between a user's elimination mechanism for alkyl halides and the solution manual's approach. The user describes using a Newman projection and neighboring group participation (NGP) to achieve anti elimination, resulting in trans-2-butene. In contrast, the solution manual employs a standard SN2 mechanism followed by iodine removal, which the user argues incorrectly yields a trans alkene instead of the expected cis alkene. The user seeks clarification on whether their understanding or the solution manual's mechanism is flawed. The conversation highlights the complexities of elimination reactions and the importance of accurate mechanisms in organic chemistry.
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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.
Screenshot_2024-09-30-14-31-44-14_f9ee0578fe1cc94de7482bd41accb329.jpg
The above is the question.
Screenshot_2024-09-30-14-46-12-11_92460851df6f172a4592fca41cc2d2e6.jpg

This is the solution provided.
 

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