A special instance of sp applied to an equality with a disjoint variable condition. Unlike the more general sp, we can prove this without [ax-12]. Instance of aeveq.

The antecedent \( \forall x \, x = y' \) with distinct \( x \) and \( y' \) is a characteristic of a degenerate universe, in which just one object exists. Actually more than one object may still exist, but if so, we give up on equality as a discriminating term.

Separating this degenerate case from a richer universe, where inequality is possible, is a common proof idea. The name of this theorem follows a convention, where the condition \( \forall x \, x = y' \) is denoted by 'aev', a shorthand for 'all equal, with a distinct variable condition'. (Contributed by Wolf Lammen, 14-Mar-2021.)