Description: Existential introduction, using implicit substitution. Compare Lemma 14 of Tarski p. 70. (Contributed by NM, 7-Aug-1994) (Proof shortened by Wolf Lammen, 22-Oct-2023)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | spimew.1 | |- ( ph -> A. x ph ) |
|
| spimew.2 | |- ( x = y -> ( ph -> ps ) ) |
||
| Assertion | spimew | |- ( ph -> E. x ps ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | spimew.1 | |- ( ph -> A. x ph ) |
|
| 2 | spimew.2 | |- ( x = y -> ( ph -> ps ) ) |
|
| 3 | ax6v | |- -. A. x -. x = y |
|
| 4 | 2 | speimfw | |- ( -. A. x -. x = y -> ( A. x ph -> E. x ps ) ) |
| 5 | 3 1 4 | mpsyl | |- ( ph -> E. x ps ) |