Description: An equivalence related to implicit substitution. Version of equsex with a disjoint variable condition, which does not require ax-13 . See equsexvw for a version with two disjoint variable conditions requiring fewer axioms. See also the dual form equsalv . (Contributed by NM, 5-Aug-1993) (Revised by BJ, 31-May-2019) Avoid ax-10 . (Revised by Gino Giotto, 18-Nov-2024)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | equsalv.nf | |- F/ x ps |
|
| equsalv.1 | |- ( x = y -> ( ph <-> ps ) ) |
||
| Assertion | equsexv | |- ( E. x ( x = y /\ ph ) <-> ps ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | equsalv.nf | |- F/ x ps |
|
| 2 | equsalv.1 | |- ( x = y -> ( ph <-> ps ) ) |
|
| 3 | 2 | biimpa | |- ( ( x = y /\ ph ) -> ps ) |
| 4 | 1 3 | exlimi | |- ( E. x ( x = y /\ ph ) -> ps ) |
| 5 | 1 2 | equsalv | |- ( A. x ( x = y -> ph ) <-> ps ) |
| 6 | equs4v | |- ( A. x ( x = y -> ph ) -> E. x ( x = y /\ ph ) ) |
|
| 7 | 5 6 | sylbir | |- ( ps -> E. x ( x = y /\ ph ) ) |
| 8 | 4 7 | impbii | |- ( E. x ( x = y /\ ph ) <-> ps ) |