Metamath Proof Explorer


Theorem ralnex2

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 ( ∀ 𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ¬ ∃ 𝑥𝐴𝑦𝐵 𝜑 )

Proof

Step Hyp Ref Expression
1 ralnex ( ∀ 𝑦𝐵 ¬ 𝜑 ↔ ¬ ∃ 𝑦𝐵 𝜑 )
2 1 ralbii ( ∀ 𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ∀ 𝑥𝐴 ¬ ∃ 𝑦𝐵 𝜑 )
3 ralnex ( ∀ 𝑥𝐴 ¬ ∃ 𝑦𝐵 𝜑 ↔ ¬ ∃ 𝑥𝐴𝑦𝐵 𝜑 )
4 2 3 bitri ( ∀ 𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ¬ ∃ 𝑥𝐴𝑦𝐵 𝜑 )