Description: Obsolete version of ifpdfbi as of 25-Jun-2026. Define the biconditional as conditional logic operator. (Contributed by RP, 20-Apr-2020) (Proof shortened by Wolf Lammen, 30-Apr-2024) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | ifpdfbiOLD | |- ( ( ph <-> ps ) <-> if- ( ph , ps , -. 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 | dfifp2 | |- ( if- ( ph , ps , -. ps ) <-> ( ( ph -> ps ) /\ ( -. ph -> -. ps ) ) ) |
|
| 5 | 2 3 4 | 3bitr4i | |- ( ( ph <-> ps ) <-> if- ( ph , ps , -. ps ) ) |