Description: An equivalence related to implicit substitution. Version of equsexh with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 5-Aug-1993) (Revised by BJ, 31-May-2019)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | equsalhw.1 | |- ( ps -> A. x ps ) |
|
| equsalhw.2 | |- ( x = y -> ( ph <-> ps ) ) |
||
| Assertion | equsexhv | |- ( E. x ( x = y /\ ph ) <-> ps ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | equsalhw.1 | |- ( ps -> A. x ps ) |
|
| 2 | equsalhw.2 | |- ( x = y -> ( ph <-> ps ) ) |
|
| 3 | 1 | nf5i | |- F/ x ps |
| 4 | 3 2 | equsexv | |- ( E. x ( x = y /\ ph ) <-> ps ) |