Description: An equivalence between an implication with a universally quantified consequent and an implication with an existentially quantified antecedent. 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 eximal . (Contributed by BJ, 12-May-2019)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | alimex | |- ( ( ph -> A. x ps ) <-> ( E. x -. ps -> -. ph ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | alex | |- ( A. x ps <-> -. E. x -. ps ) |
|
| 2 | 1 | imbi2i | |- ( ( ph -> A. x ps ) <-> ( ph -> -. E. x -. ps ) ) |
| 3 | con2b | |- ( ( ph -> -. E. x -. ps ) <-> ( E. x -. ps -> -. ph ) ) |
|
| 4 | 2 3 | bitri | |- ( ( ph -> A. x ps ) <-> ( E. x -. ps -> -. ph ) ) |