Description: Theorem 19.38 of Margaris p. 90. The converse holds under nonfreeness conditions, see 19.38a and 19.38b . (Contributed by NM, 12-Mar-1993) Allow a shortening of 19.21t . (Revised by Wolf Lammen, 2-Jan-2018)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | 19.38 | |- ( ( E. x ph -> A. x ps ) -> A. x ( ph -> ps ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | alnex | |- ( A. x -. ph <-> -. E. x ph ) |
|
| 2 | pm2.21 | |- ( -. ph -> ( ph -> ps ) ) |
|
| 3 | 2 | alimi | |- ( A. x -. ph -> A. x ( ph -> ps ) ) |
| 4 | 1 3 | sylbir | |- ( -. E. x ph -> A. x ( ph -> ps ) ) |
| 5 | ala1 | |- ( A. x ps -> A. x ( ph -> ps ) ) |
|
| 6 | 4 5 | ja | |- ( ( E. x ph -> A. x ps ) -> A. x ( ph -> ps ) ) |