Description: Double existential uniqueness implies double unique existential quantification. The converse does not hold. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker 2exeuv when possible. (Contributed by NM, 3-Dec-2001) (Proof shortened by Mario Carneiro, 22-Dec-2016) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | 2exeu | |- ( ( E! x E. y ph /\ E! y E. x ph ) -> E! x E! y ph ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eumo | |- ( E! x E. y ph -> E* x E. y ph ) |
|
| 2 | euex | |- ( E! y ph -> E. y ph ) |
|
| 3 | 2 | moimi | |- ( E* x E. y ph -> E* x E! y ph ) |
| 4 | 1 3 | syl | |- ( E! x E. y ph -> E* x E! y ph ) |
| 5 | 2euex | |- ( E! y E. x ph -> E. x E! y ph ) |
|
| 6 | 4 5 | anim12ci | |- ( ( E! x E. y ph /\ E! y E. x ph ) -> ( E. x E! y ph /\ E* x E! y ph ) ) |
| 7 | df-eu | |- ( E! x E! y ph <-> ( E. x E! y ph /\ E* x E! y ph ) ) |
|
| 8 | 6 7 | sylibr | |- ( ( E! x E. y ph /\ E! y E. x ph ) -> E! x E! y ph ) |