Description: Equality theorem for restricted at-most-one quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017) Remove usage of ax-10 , ax-11 , and ax-12 . (Revised by Steven Nguyen, 30-Apr-2023)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | rmoeq1 | |- ( 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 | mobidv | |- ( A = B -> ( E* x ( x e. A /\ ph ) <-> E* x ( x e. B /\ ph ) ) ) |
| 4 | df-rmo | |- ( E* x e. A ph <-> E* x ( x e. A /\ ph ) ) |
|
| 5 | df-rmo | |- ( 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 ) ) |