Description: Selecting one of two alternatives within a restricted generalization if one of the alternatives is false. (Contributed by AV, 6-Sep-2018) (Proof shortened by AV, 13-Oct-2018)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | ralralimp | |- ( ( ph /\ A =/= (/) ) -> ( A. x e. A ( ( ph -> ( th \/ ta ) ) /\ -. th ) -> ta ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ornld | |- ( ph -> ( ( ( ph -> ( th \/ ta ) ) /\ -. th ) -> ta ) ) |
|
| 2 | 1 | adantr | |- ( ( ph /\ A =/= (/) ) -> ( ( ( ph -> ( th \/ ta ) ) /\ -. th ) -> ta ) ) |
| 3 | 2 | ralimdv | |- ( ( ph /\ A =/= (/) ) -> ( A. x e. A ( ( ph -> ( th \/ ta ) ) /\ -. th ) -> A. x e. A ta ) ) |
| 4 | rspn0 | |- ( A =/= (/) -> ( A. x e. A ta -> ta ) ) |
|
| 5 | 4 | adantl | |- ( ( ph /\ A =/= (/) ) -> ( A. x e. A ta -> ta ) ) |
| 6 | 3 5 | syld | |- ( ( ph /\ A =/= (/) ) -> ( A. x e. A ( ( ph -> ( th \/ ta ) ) /\ -. th ) -> ta ) ) |