Description: Relationship between two restricted universal and existential quantifiers. (Contributed by Glauco Siliprandi, 11-Dec-2019) (Proof shortened by Wolf Lammen, 18-May-2023)
Ref | Expression | ||
---|---|---|---|
Assertion | ralnex2 | ⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralnex | ⊢ ( ∀ 𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∃ 𝑦 ∈ 𝐵 𝜑 ) | |
2 | 1 | ralbii | ⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 ¬ ∃ 𝑦 ∈ 𝐵 𝜑 ) |
3 | ralnex | ⊢ ( ∀ 𝑥 ∈ 𝐴 ¬ ∃ 𝑦 ∈ 𝐵 𝜑 ↔ ¬ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 ) | |
4 | 2 3 | bitri | ⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 ) |