Verifying a subtlety of Karp's original proof that SAT has a polynomial time reduction to 3SAT
The conjunction of the first two clauses, $(\sigma_1\cup\sigma_2\cup u_1)(\sigma_3\cup\ldots\cup\sigma_m\cup\bar{u}_1)$ is equisatisfiable to the original clause, as can be easily checked (any valuation satisfying one of the $\sigma_i$ also satisfies one of the two new clauses, and can be extended to $u_1$ as needed to satisfy the other; any valuation satisfying both new clauses has to satisfy some $\sigma_i$, since it cannot satisfy both $u_1$ and $\bar{u}_1$.) 

The additional clauses ensure that $u_1$ is in fact equivalent to $\sigma_3\cup\ldots\cup\sigma_m$, but as far as I can see, Karp does not actually use this. He credits Cook's 1971 paper for this particular reduction, but the version found there (actually for DNF tautologies rather than CNF satisfiability) does not feature these extra constraints.