Description: Introduce unique existential quantifier to both sides of an equivalence. (Contributed by NM, 9-Jul-1994) (Revised by Mario Carneiro, 6-Oct-2016) Avoid ax-5 . (Revised by Wolf Lammen, 27-Sep-2023)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | eubii.1 | |- ( ph <-> ps ) |
|
| Assertion | eubii | |- ( E! x ph <-> E! x ps ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eubii.1 | |- ( ph <-> ps ) |
|
| 2 | 1 | exbii | |- ( E. x ph <-> E. x ps ) |
| 3 | 1 | mobii | |- ( E* x ph <-> E* x ps ) |
| 4 | 2 3 | anbi12i | |- ( ( E. x ph /\ E* x ph ) <-> ( E. x ps /\ E* x ps ) ) |
| 5 | df-eu | |- ( E! x ph <-> ( E. x ph /\ E* x ph ) ) |
|
| 6 | df-eu | |- ( E! x ps <-> ( E. x ps /\ E* x ps ) ) |
|
| 7 | 4 5 6 | 3bitr4i | |- ( E! x ph <-> E! x ps ) |