Description: An equivalence between an implication with an existentially quantified antecedent and an implication with a universally quantified consequent. An interesting case is when the same formula is substituted for both ph and ps , since then both implications express a type of non-freeness. See also alimex . (Contributed by BJ, 12-May-2019)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | eximal | |- ( ( E. x ph -> ps ) <-> ( -. ps -> A. x -. ph ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-ex | |- ( E. x ph <-> -. A. x -. ph ) |
|
| 2 | 1 | imbi1i | |- ( ( E. x ph -> ps ) <-> ( -. A. x -. ph -> ps ) ) |
| 3 | con1b | |- ( ( -. A. x -. ph -> ps ) <-> ( -. ps -> A. x -. ph ) ) |
|
| 4 | 2 3 | bitri | |- ( ( E. x ph -> ps ) <-> ( -. ps -> A. x -. ph ) ) |