Description: Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997) (Proof shortened by BJ, 16-Jul-2021)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | ralnex | |- ( A. x e. A -. ph <-> -. E. x e. A ph ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | raln | |- ( A. x e. A -. ph <-> A. x -. ( x e. A /\ ph ) ) |
|
| 2 | alnex | |- ( A. x -. ( x e. A /\ ph ) <-> -. E. x ( x e. A /\ ph ) ) |
|
| 3 | df-rex | |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) ) |
|
| 4 | 2 3 | xchbinxr | |- ( A. x -. ( x e. A /\ ph ) <-> -. E. x e. A ph ) |
| 5 | 1 4 | bitri | |- ( A. x e. A -. ph <-> -. E. x e. A ph ) |