Description: Existential uniqueness implies there is a value for which the wff argument is false. (Contributed by NM, 24-Oct-2010) (Proof shortened by BJ, 2-Jan-2023)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | eunex | |- ( E! x ph -> E. x -. ph ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dtru | |- -. A. x x = y |
|
| 2 | albi | |- ( A. x ( ph <-> x = y ) -> ( A. x ph <-> A. x x = y ) ) |
|
| 3 | 1 2 | mtbiri | |- ( A. x ( ph <-> x = y ) -> -. A. x ph ) |
| 4 | 3 | exlimiv | |- ( E. y A. x ( ph <-> x = y ) -> -. A. x ph ) |
| 5 | eu6 | |- ( E! x ph <-> E. y A. x ( ph <-> x = y ) ) |
|
| 6 | exnal | |- ( E. x -. ph <-> -. A. x ph ) |
|
| 7 | 4 5 6 | 3imtr4i | |- ( E! x ph -> E. x -. ph ) |