Description: A special instance of sp 2167
applied to an equality with a disjoint
variable condition. Unlike the more general sp 2167, we
can prove this
without ax-12 2163. Instance of aeveq 2099.
The antecedent ∀𝑥𝑥 = 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 ∀𝑥𝑥 = 𝑦 is denoted by 'aev', a
shorthand for 'all equal, with a distinct variable condition'.
(Contributed by Wolf Lammen, 14-Mar-2021.) |