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Metamath Proof Explorer

Theorem 2exanali

Description: Theorem *11.521 in WhiteheadRussell p. 164. (Contributed by Andrew Salmon, 24-May-2011)

Ref Expression
Assertion 2exanali ⊢ ¬ ∃ x ∃ y φ ∧ ¬ ψ ↔ ∀ x ∀ y φ → ψ

Proof

Step Hyp Ref Expression
1 2nalexn ⊢ ¬ ∀ x ∀ y φ → ψ ↔ ∃ x ∃ y ¬ φ → ψ
2 1 con1bii ⊢ ¬ ∃ x ∃ y ¬ φ → ψ ↔ ∀ x ∀ y φ → ψ
3 annim ⊢ φ ∧ ¬ ψ ↔ ¬ φ → ψ
4 3 2exbii ⊢ ∃ x ∃ y φ ∧ ¬ ψ ↔ ∃ x ∃ y ¬ φ → ψ
5 2 4 xchnxbir ⊢ ¬ ∃ x ∃ y φ ∧ ¬ ψ ↔ ∀ x ∀ y φ → ψ