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)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | reueq1 | |- ( A = B -> ( E! x e. A ph <-> E! x e. B ph ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eleq2 | |- ( A = B -> ( x e. A <-> x e. B ) ) |
|
| 2 | 1 | anbi1d | |- ( A = B -> ( ( x e. A /\ ph ) <-> ( x e. B /\ ph ) ) ) |
| 3 | 2 | eubidv | |- ( A = B -> ( E! x ( x e. A /\ ph ) <-> E! x ( x e. B /\ ph ) ) ) |
| 4 | df-reu | |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) ) |
|
| 5 | df-reu | |- ( E! x e. B ph <-> E! x ( x e. B /\ ph ) ) |
|
| 6 | 3 4 5 | 3bitr4g | |- ( A = B -> ( E! x e. A ph <-> E! x e. B ph ) ) |