Metamath Proof Explorer


Theorem nrexralim

Description: Negation of a complex predicate calculus formula. (Contributed by FL, 31-Jul-2009)

Ref Expression
Assertion nrexralim ( ¬ ∃ 𝑥𝐴𝑦𝐵 ( 𝜑𝜓 ) ↔ ∀ 𝑥𝐴𝑦𝐵 ( 𝜑 ∧ ¬ 𝜓 ) )

Proof

Step Hyp Ref Expression
1 rexanali ( ∃ 𝑦𝐵 ( 𝜑 ∧ ¬ 𝜓 ) ↔ ¬ ∀ 𝑦𝐵 ( 𝜑𝜓 ) )
2 1 ralbii ( ∀ 𝑥𝐴𝑦𝐵 ( 𝜑 ∧ ¬ 𝜓 ) ↔ ∀ 𝑥𝐴 ¬ ∀ 𝑦𝐵 ( 𝜑𝜓 ) )
3 ralnex ( ∀ 𝑥𝐴 ¬ ∀ 𝑦𝐵 ( 𝜑𝜓 ) ↔ ¬ ∃ 𝑥𝐴𝑦𝐵 ( 𝜑𝜓 ) )
4 2 3 bitr2i ( ¬ ∃ 𝑥𝐴𝑦𝐵 ( 𝜑𝜓 ) ↔ ∀ 𝑥𝐴𝑦𝐵 ( 𝜑 ∧ ¬ 𝜓 ) )