Description: Relationship between three restricted universal and existential quantifiers. (Contributed by Thierry Arnoux, 12-Jul-2020) (Proof shortened by Wolf Lammen, 18-May-2023)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | ralnex3 | ⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 ¬ 𝜑 ↔ ¬ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ∃ 𝑧 ∈ 𝐶 𝜑 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ralnex | ⊢ ( ∀ 𝑧 ∈ 𝐶 ¬ 𝜑 ↔ ¬ ∃ 𝑧 ∈ 𝐶 𝜑 ) | |
| 2 | 1 | 2ralbii | ⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 ¬ 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ¬ ∃ 𝑧 ∈ 𝐶 𝜑 ) |
| 3 | ralnex2 | ⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ¬ ∃ 𝑧 ∈ 𝐶 𝜑 ↔ ¬ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ∃ 𝑧 ∈ 𝐶 𝜑 ) | |
| 4 | 2 3 | bitri | ⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 ¬ 𝜑 ↔ ¬ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ∃ 𝑧 ∈ 𝐶 𝜑 ) |