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 | |- ( A. x x = y -> x = y ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | equequ1 | |- ( x = z -> ( x = y <-> z = y ) ) |
|
| 2 | 1 | spw | |- ( A. x x = y -> x = y ) |