Theorem 2nexaln

Description: Theorem *11.25 in WhiteheadRussell p. 160. (Contributed by Andrew Salmon, 24-May-2011)

	Ref	Expression
Assertion	2nexaln	⊢ ( ¬ ∃ 𝑥 ∃ 𝑦 𝜑 ↔ ∀ 𝑥 ∀ 𝑦 ¬ 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		2exnaln	⊢ ( ∃ 𝑥 ∃ 𝑦 𝜑 ↔ ¬ ∀ 𝑥 ∀ 𝑦 ¬ 𝜑 )
2	1	bicomi	⊢ ( ¬ ∀ 𝑥 ∀ 𝑦 ¬ 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 𝜑 )
3	2	con1bii	⊢ ( ¬ ∃ 𝑥 ∃ 𝑦 𝜑 ↔ ∀ 𝑥 ∀ 𝑦 ¬ 𝜑 )