Description: Deduction adding a consequent to both sides of a logical equivalence. (Contributed by NM, 11-May-1993) (Proof shortened by Wolf Lammen, 17-Sep-2013)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | imbid.1 | |- ( ph -> ( ps <-> ch ) ) |
|
| Assertion | imbi1d | |- ( ph -> ( ( ps -> th ) <-> ( ch -> th ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | imbid.1 | |- ( ph -> ( ps <-> ch ) ) |
|
| 2 | 1 | biimprd | |- ( ph -> ( ch -> ps ) ) |
| 3 | 2 | imim1d | |- ( ph -> ( ( ps -> th ) -> ( ch -> th ) ) ) |
| 4 | 1 | biimpd | |- ( ph -> ( ps -> ch ) ) |
| 5 | 4 | imim1d | |- ( ph -> ( ( ch -> th ) -> ( ps -> th ) ) ) |
| 6 | 3 5 | impbid | |- ( ph -> ( ( ps -> th ) <-> ( ch -> th ) ) ) |