Description: Distributive law for implication over conjunction. Compare Theorem *4.76 of WhiteheadRussell p. 121. (Contributed by NM, 3-Apr-1994) (Proof shortened by Wolf Lammen, 27-Nov-2013)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | jcab | |- ( ( ph -> ( ps /\ ch ) ) <-> ( ( ph -> ps ) /\ ( ph -> ch ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simpl | |- ( ( ps /\ ch ) -> ps ) |
|
| 2 | 1 | imim2i | |- ( ( ph -> ( ps /\ ch ) ) -> ( ph -> ps ) ) |
| 3 | simpr | |- ( ( ps /\ ch ) -> ch ) |
|
| 4 | 3 | imim2i | |- ( ( ph -> ( ps /\ ch ) ) -> ( ph -> ch ) ) |
| 5 | 2 4 | jca | |- ( ( ph -> ( ps /\ ch ) ) -> ( ( ph -> ps ) /\ ( ph -> ch ) ) ) |
| 6 | pm3.43 | |- ( ( ( ph -> ps ) /\ ( ph -> ch ) ) -> ( ph -> ( ps /\ ch ) ) ) |
|
| 7 | 5 6 | impbii | |- ( ( ph -> ( ps /\ ch ) ) <-> ( ( ph -> ps ) /\ ( ph -> ch ) ) ) |