Description: Theorem 19.22 of Margaris p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Nov-1994) (Proof shortened by Andrew Salmon, 30-May-2011)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | rexim | |- ( A. x e. A ( ph -> ps ) -> ( E. x e. A ph -> E. x e. A ps ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | con3 | |- ( ( ph -> ps ) -> ( -. ps -> -. ph ) ) |
|
| 2 | 1 | ral2imi | |- ( A. x e. A ( ph -> ps ) -> ( A. x e. A -. ps -> A. x e. A -. ph ) ) |
| 3 | ralnex | |- ( A. x e. A -. ps <-> -. E. x e. A ps ) |
|
| 4 | ralnex | |- ( A. x e. A -. ph <-> -. E. x e. A ph ) |
|
| 5 | 2 3 4 | 3imtr3g | |- ( A. x e. A ( ph -> ps ) -> ( -. E. x e. A ps -> -. E. x e. A ph ) ) |
| 6 | 5 | con4d | |- ( A. x e. A ( ph -> ps ) -> ( E. x e. A ph -> E. x e. A ps ) ) |