Description: Closed theorem form of spim . Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 15-Jan-2008) (Revised by Mario Carneiro, 17-Oct-2016) (Proof shortened by Wolf Lammen, 21-Mar-2023) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | spimt | |- ( ( F/ x ps /\ A. x ( x = y -> ( ph -> ps ) ) ) -> ( A. x ph -> ps ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ax6e | |- E. x x = y |
|
| 2 | exim | |- ( A. x ( x = y -> ( ph -> ps ) ) -> ( E. x x = y -> E. x ( ph -> ps ) ) ) |
|
| 3 | 1 2 | mpi | |- ( A. x ( x = y -> ( ph -> ps ) ) -> E. x ( ph -> ps ) ) |
| 4 | 19.35 | |- ( E. x ( ph -> ps ) <-> ( A. x ph -> E. x ps ) ) |
|
| 5 | 3 4 | sylib | |- ( A. x ( x = y -> ( ph -> ps ) ) -> ( A. x ph -> E. x ps ) ) |
| 6 | id | |- ( F/ x ps -> F/ x ps ) |
|
| 7 | 6 | 19.9d | |- ( F/ x ps -> ( E. x ps -> ps ) ) |
| 8 | 5 7 | sylan9r | |- ( ( F/ x ps /\ A. x ( x = y -> ( ph -> ps ) ) ) -> ( A. x ph -> ps ) ) |