Let $\phi(x_1,...,x_n)$ be a statement about an equality of two expressions having $x_1,...,x_n$ respectively.
If there is no $(x_1,...,x_n)$ such that $\phi(x_1,...,x_n)$ is true, we call this equation $\phi$ inconsistent. I don't know why specially 'consistency and inconsistency' is named for equations. Literally, 'An equation has no solution' makes more sense than 'an equation is inconsistent' to me. If $\phi$ is inconsistent, does this mean something about $Con(\phi)$?