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Metamath Proof Explorer

Theorem alinexa

Description: A transformation of quantifiers and logical connectives. (Contributed by NM, 19-Aug-1993)

Ref Expression
Assertion alinexa ⊢ ∀ x φ → ¬ ψ ↔ ¬ ∃ x φ ∧ ψ

Proof

Step Hyp Ref Expression
1 imnang ⊢ ∀ x φ → ¬ ψ ↔ ∀ x ¬ φ ∧ ψ
2 alnex ⊢ ∀ x ¬ φ ∧ ψ ↔ ¬ ∃ x φ ∧ ψ
3 1 2 bitri ⊢ ∀ x φ → ¬ ψ ↔ ¬ ∃ x φ ∧ ψ