Theorem 2nalexn

Description: Part of theorem *11.5 in WhiteheadRussell p. 164. (Contributed by Andrew Salmon, 24-May-2011)

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

Proof

Step	Hyp	Ref	Expression
1		df-ex	⊢ ( ∃ 𝑥 ∃ 𝑦 ¬ 𝜑 ↔ ¬ ∀ 𝑥 ¬ ∃ 𝑦 ¬ 𝜑 )
2		alex	⊢ ( ∀ 𝑦 𝜑 ↔ ¬ ∃ 𝑦 ¬ 𝜑 )
3	2	albii	⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 ↔ ∀ 𝑥 ¬ ∃ 𝑦 ¬ 𝜑 )
4	1 3	xchbinxr	⊢ ( ∃ 𝑥 ∃ 𝑦 ¬ 𝜑 ↔ ¬ ∀ 𝑥 ∀ 𝑦 𝜑 )
5	4	bicomi	⊢ ( ¬ ∀ 𝑥 ∀ 𝑦 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 ¬ 𝜑 )