Description: Negated triple conjunction expressed in terms of triple disjunction. (Contributed by Jeff Hankins, 15-Aug-2009) (Proof shortened by Andrew Salmon, 13-May-2011) (Revised by Wolf Lammen, 8-Apr-2022)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | 3ianor | ⊢ ( ¬ ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ( ¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ianor | ⊢ ( ¬ ( 𝜑 ∧ 𝜓 ) ↔ ( ¬ 𝜑 ∨ ¬ 𝜓 ) ) | |
| 2 | 1 | orbi1i | ⊢ ( ( ¬ ( 𝜑 ∧ 𝜓 ) ∨ ¬ 𝜒 ) ↔ ( ( ¬ 𝜑 ∨ ¬ 𝜓 ) ∨ ¬ 𝜒 ) ) |
| 3 | ianor | ⊢ ( ¬ ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) ↔ ( ¬ ( 𝜑 ∧ 𝜓 ) ∨ ¬ 𝜒 ) ) | |
| 4 | df-3an | ⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) ) | |
| 5 | 3 4 | xchnxbir | ⊢ ( ¬ ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ( ¬ ( 𝜑 ∧ 𝜓 ) ∨ ¬ 𝜒 ) ) |
| 6 | df-3or | ⊢ ( ( ¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒 ) ↔ ( ( ¬ 𝜑 ∨ ¬ 𝜓 ) ∨ ¬ 𝜒 ) ) | |
| 7 | 2 5 6 | 3bitr4i | ⊢ ( ¬ ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ( ¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒 ) ) |