How to prove over first order logic that if B has free variables $\{x_1, x_2\}$ it keeps the dichotomy theorem. namely: $$if\ \Gamma, A \vDash_v B ,\Gamma, \neg A \vDash_v B$$ $$ \ then \ \Gamma, \vDash_v B $$
where:
$\Gamma$ is a group of statements, A and B are statements.
$\vDash_v$ is means that a formula is satisfied if and only if its universal closure is satisfied. or that B is satisfied by M for every assignment of $\{x_1, x_2\}$.