Theorem nrexralim

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

	Ref	Expression
Assertion	nrexralim	⊢ ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜑 → 𝜓 ) ↔ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ( 𝜑 ∧ ¬ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		rexanali	⊢ ( ∃ 𝑦 ∈ 𝐵 ( 𝜑 ∧ ¬ 𝜓 ) ↔ ¬ ∀ 𝑦 ∈ 𝐵 ( 𝜑 → 𝜓 ) )
2	1	ralbii	⊢ ( ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ( 𝜑 ∧ ¬ 𝜓 ) ↔ ∀ 𝑥 ∈ 𝐴 ¬ ∀ 𝑦 ∈ 𝐵 ( 𝜑 → 𝜓 ) )
3		ralnex	⊢ ( ∀ 𝑥 ∈ 𝐴 ¬ ∀ 𝑦 ∈ 𝐵 ( 𝜑 → 𝜓 ) ↔ ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜑 → 𝜓 ) )
4	2 3	bitr2i	⊢ ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝜑 → 𝜓 ) ↔ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ( 𝜑 ∧ ¬ 𝜓 ) )