Description: Deduction adding a biconditional to the left in an equivalence. (Contributed by NM, 11-May-1993) (Proof shortened by Wolf Lammen, 19-May-2013)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | imbid.1 | |- ( ph -> ( ps <-> ch ) ) |
|
| Assertion | bibi2d | |- ( ph -> ( ( th <-> ps ) <-> ( th <-> ch ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | imbid.1 | |- ( ph -> ( ps <-> ch ) ) |
|
| 2 | 1 | pm5.74i | |- ( ( ph -> ps ) <-> ( ph -> ch ) ) |
| 3 | 2 | bibi2i | |- ( ( ( ph -> th ) <-> ( ph -> ps ) ) <-> ( ( ph -> th ) <-> ( ph -> ch ) ) ) |
| 4 | pm5.74 | |- ( ( ph -> ( th <-> ps ) ) <-> ( ( ph -> th ) <-> ( ph -> ps ) ) ) |
|
| 5 | pm5.74 | |- ( ( ph -> ( th <-> ch ) ) <-> ( ( ph -> th ) <-> ( ph -> ch ) ) ) |
|
| 6 | 3 4 5 | 3bitr4i | |- ( ( ph -> ( th <-> ps ) ) <-> ( ph -> ( th <-> ch ) ) ) |
| 7 | 6 | pm5.74ri | |- ( ph -> ( ( th <-> ps ) <-> ( th <-> ch ) ) ) |