Description: Version of dvelim without any variable restrictions. Usage of this theorem is discouraged because it depends on ax-13 . Check out dvelimhw for a version requiring fewer axioms. (Contributed by NM, 1-Oct-2002) (Proof shortened by Wolf Lammen, 11-May-2018) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | dvelimh.1 | |- ( ph -> A. x ph ) |
|
| dvelimh.2 | |- ( ps -> A. z ps ) |
||
| dvelimh.3 | |- ( z = y -> ( ph <-> ps ) ) |
||
| Assertion | dvelimh | |- ( -. A. x x = y -> ( ps -> A. x ps ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dvelimh.1 | |- ( ph -> A. x ph ) |
|
| 2 | dvelimh.2 | |- ( ps -> A. z ps ) |
|
| 3 | dvelimh.3 | |- ( z = y -> ( ph <-> ps ) ) |
|
| 4 | 1 | nf5i | |- F/ x ph |
| 5 | 2 | nf5i | |- F/ z ps |
| 6 | 4 5 3 | dvelimf | |- ( -. A. x x = y -> F/ x ps ) |
| 7 | 6 | nf5rd | |- ( -. A. x x = y -> ( ps -> A. x ps ) ) |