Description: Disjunction of antecedents. Compare Theorem *4.77 of WhiteheadRussell p. 121. (Contributed by NM, 30-May-1994) (Proof shortened by Wolf Lammen, 9-Dec-2012)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | jaob | ⊢ ( ( ( 𝜑 ∨ 𝜒 ) → 𝜓 ) ↔ ( ( 𝜑 → 𝜓 ) ∧ ( 𝜒 → 𝜓 ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | pm2.67-2 | ⊢ ( ( ( 𝜑 ∨ 𝜒 ) → 𝜓 ) → ( 𝜑 → 𝜓 ) ) | |
| 2 | olc | ⊢ ( 𝜒 → ( 𝜑 ∨ 𝜒 ) ) | |
| 3 | 2 | imim1i | ⊢ ( ( ( 𝜑 ∨ 𝜒 ) → 𝜓 ) → ( 𝜒 → 𝜓 ) ) |
| 4 | 1 3 | jca | ⊢ ( ( ( 𝜑 ∨ 𝜒 ) → 𝜓 ) → ( ( 𝜑 → 𝜓 ) ∧ ( 𝜒 → 𝜓 ) ) ) |
| 5 | pm3.44 | ⊢ ( ( ( 𝜑 → 𝜓 ) ∧ ( 𝜒 → 𝜓 ) ) → ( ( 𝜑 ∨ 𝜒 ) → 𝜓 ) ) | |
| 6 | 4 5 | impbii | ⊢ ( ( ( 𝜑 ∨ 𝜒 ) → 𝜓 ) ↔ ( ( 𝜑 → 𝜓 ) ∧ ( 𝜒 → 𝜓 ) ) ) |