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Metamath Proof Explorer

Theorem 2exnexn

Description: Theorem *11.51 in WhiteheadRussell p. 164. (Contributed by Andrew Salmon, 24-May-2011) (Proof shortened by Wolf Lammen, 25-Sep-2014)

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

Proof

Step Hyp Ref Expression
1 alexn ⊢ ( ∀ 𝑥 ∃ 𝑦 ¬ 𝜑 ↔ ¬ ∃ 𝑥 ∀ 𝑦 𝜑 )
2 1 con2bii ⊢ ( ∃ 𝑥 ∀ 𝑦 𝜑 ↔ ¬ ∀ 𝑥 ∃ 𝑦 ¬ 𝜑 )