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 | ⊢ ( ( 𝜑 → ( 𝜓 ∧ 𝜒 ) ) ↔ ( ( 𝜑 → 𝜓 ) ∧ ( 𝜑 → 𝜒 ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simpl | ⊢ ( ( 𝜓 ∧ 𝜒 ) → 𝜓 ) | |
| 2 | 1 | imim2i | ⊢ ( ( 𝜑 → ( 𝜓 ∧ 𝜒 ) ) → ( 𝜑 → 𝜓 ) ) |
| 3 | simpr | ⊢ ( ( 𝜓 ∧ 𝜒 ) → 𝜒 ) | |
| 4 | 3 | imim2i | ⊢ ( ( 𝜑 → ( 𝜓 ∧ 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) |
| 5 | 2 4 | jca | ⊢ ( ( 𝜑 → ( 𝜓 ∧ 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) ∧ ( 𝜑 → 𝜒 ) ) ) |
| 6 | pm3.43 | ⊢ ( ( ( 𝜑 → 𝜓 ) ∧ ( 𝜑 → 𝜒 ) ) → ( 𝜑 → ( 𝜓 ∧ 𝜒 ) ) ) | |
| 7 | 5 6 | impbii | ⊢ ( ( 𝜑 → ( 𝜓 ∧ 𝜒 ) ) ↔ ( ( 𝜑 → 𝜓 ) ∧ ( 𝜑 → 𝜒 ) ) ) |