Description: Equality theorem for restricted unique existential quantifier. (Contributed by NM, 5-Apr-2004) Remove usage of ax-10 , ax-11 , and ax-12 . (Revised by Steven Nguyen, 30-Apr-2023) Avoid ax-8 . (Revised by Wolf Lammen, 12-Mar-2025)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | reueq1 | |- ( A = B -> ( E! x e. A ph <-> E! x e. B ph ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rexeq | |- ( A = B -> ( E. x e. A ph <-> E. x e. B ph ) ) |
|
| 2 | rmoeq1 | |- ( A = B -> ( E* x e. A ph <-> E* x e. B ph ) ) |
|
| 3 | 1 2 | anbi12d | |- ( A = B -> ( ( E. x e. A ph /\ E* x e. A ph ) <-> ( E. x e. B ph /\ E* x e. B ph ) ) ) |
| 4 | reu5 | |- ( E! x e. A ph <-> ( E. x e. A ph /\ E* x e. A ph ) ) |
|
| 5 | reu5 | |- ( E! x e. B ph <-> ( E. x e. B ph /\ E* x e. B ph ) ) |
|
| 6 | 3 4 5 | 3bitr4g | |- ( A = B -> ( E! x e. A ph <-> E! x e. B ph ) ) |