Description: The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005) (Proof shortened by JJ, 14-Jul-2021)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | ralf0.1 | |- -. ph |
|
| Assertion | ralf0 | |- ( A. x e. A ph <-> A = (/) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ralf0.1 | |- -. ph |
|
| 2 | mtt | |- ( -. ph -> ( -. x e. A <-> ( x e. A -> ph ) ) ) |
|
| 3 | 1 2 | ax-mp | |- ( -. x e. A <-> ( x e. A -> ph ) ) |
| 4 | 3 | albii | |- ( A. x -. x e. A <-> A. x ( x e. A -> ph ) ) |
| 5 | eq0 | |- ( A = (/) <-> A. x -. x e. A ) |
|
| 6 | df-ral | |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) ) |
|
| 7 | 4 5 6 | 3bitr4ri | |- ( A. x e. A ph <-> A = (/) ) |