Metamath Proof Explorer

Theorem spaev

Description: 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 A. 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 A. x x = y is denoted by 'aev', a shorthand for 'all equal, with a distinct variable condition'. (Contributed by Wolf Lammen, 14-Mar-2021)

Ref Expression
Assertion spaev ${⊢}\forall {x}\phantom{\rule{.4em}{0ex}}{x}={y}\to {x}={y}$

Proof

Step Hyp Ref Expression
1 equequ1 ${⊢}{x}={z}\to \left({x}={y}↔{z}={y}\right)$
2 1 spw ${⊢}\forall {x}\phantom{\rule{.4em}{0ex}}{x}={y}\to {x}={y}$