Description: Version of equsexv with a disjoint variable condition, and of equsex with two disjoint variable conditions, which requires fewer axioms. See also the dual form equsalvw . (Contributed by BJ, 31-May-2019) (Proof shortened by Wolf Lammen, 23-Oct-2023)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | equsalvw.1 | |- ( x = y -> ( ph <-> ps ) ) |
|
| Assertion | equsexvw | |- ( E. x ( x = y /\ ph ) <-> ps ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | equsalvw.1 | |- ( x = y -> ( ph <-> ps ) ) |
|
| 2 | alinexa | |- ( A. x ( x = y -> -. ph ) <-> -. E. x ( x = y /\ ph ) ) |
|
| 3 | 1 | notbid | |- ( x = y -> ( -. ph <-> -. ps ) ) |
| 4 | 3 | equsalvw | |- ( A. x ( x = y -> -. ph ) <-> -. ps ) |
| 5 | 2 4 | bitr3i | |- ( -. E. x ( x = y /\ ph ) <-> -. ps ) |
| 6 | 5 | con4bii | |- ( E. x ( x = y /\ ph ) <-> ps ) |