Description: An alternate definition of the biconditional. Theorem *5.23 of WhiteheadRussell p. 124. (Contributed by NM, 27-Jun-2002) (Proof shortened by Wolf Lammen, 3-Nov-2013) (Proof shortened by NM, 29-Oct-2021)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | dfbi3 | |- ( ( ph <-> ps ) <-> ( ( ph /\ ps ) \/ ( -. ph /\ -. ps ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | con34b | |- ( ( ps -> ph ) <-> ( -. ph -> -. ps ) ) |
|
| 2 | 1 | anbi2i | |- ( ( ( ph -> ps ) /\ ( ps -> ph ) ) <-> ( ( ph -> ps ) /\ ( -. ph -> -. ps ) ) ) |
| 3 | dfbi2 | |- ( ( ph <-> ps ) <-> ( ( ph -> ps ) /\ ( ps -> ph ) ) ) |
|
| 4 | cases2 | |- ( ( ( ph /\ ps ) \/ ( -. ph /\ -. ps ) ) <-> ( ( ph -> ps ) /\ ( -. ph -> -. ps ) ) ) |
|
| 5 | 2 3 4 | 3bitr4i | |- ( ( ph <-> ps ) <-> ( ( ph /\ ps ) \/ ( -. ph /\ -. ps ) ) ) |