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In 2004 Atserias, Kolaitis, and Vardi proposed 
$\text {OBDD}$-based propositional proof systems that prove unsatisfiability of a CNF formula by deduction of an identically false 
$\text {OBDD}$ from 
$\text {OBDD}$s representing clauses of the initial formula. All 
$\text {OBDD}$s in such proofs have the same order of variables. We initiate the study of 
$\text {OBDD}$ based proof systems that additionally contain a rule that allows changing the order in 
$\text {OBDD}$s. At first we consider a proof system 
$\text {OBDD}(\land , \text{reordering})$ that uses the conjunction (join) rule and the rule that allows changing the order. We exponentially separate this proof system from 
$\text {OBDD}(\land )$ proof system that uses only the conjunction rule. We prove exponential lower bounds on the size of 
$\text {OBDD}(\land , \text{reordering})$ refutations of Tseitin formulas and the pigeonhole principle. The first lower bound was previously unknown even for 
$\text {OBDD}(\land )$ proofs and the second one extends the result of Tveretina et al. from 
$\text {OBDD}(\land )$ to 
$\text {OBDD}(\land , \text{reordering})$.
In 2001 Aguirre and Vardi proposed an approach to the propositional satisfiability problem based on 
$\text {OBDD}$s and symbolic quantifier elimination (we denote algorithms based on this approach as 
$\text {OBDD}(\land , \exists )$ algorithms). We augment these algorithms with the operation of reordering of variables and call the new scheme 
$\text {OBDD}(\land , \exists , \text{reordering})$ algorithms. We notice that there exists an 
$\text {OBDD}(\land , \exists )$ algorithm that solves satisfiable and unsatisfiable Tseitin formulas in polynomial time (a standard example of a hard system of linear equations over 
$\mathbb {F}_2$), but we show that there are formulas representing systems of linear equations over 
$\mathbb {F}_2$ that are hard for 
$\text {OBDD}(\land , \exists , \text{reordering})$ algorithms. Our hard instances are satisfiable formulas representing systems of linear equations over 
$\mathbb {F}_2$ that correspond to checksum matrices of error correcting codes.
