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Metamath Proof Explorer

Theorem nanim

Description: Implication in terms of alternative denial. (Contributed by Jeff Hoffman, 19-Nov-2007)

Ref Expression
Assertion nanim ⊢ φ → ψ ↔ φ ⊼ ψ ⊼ ψ

Proof

Step Hyp Ref Expression
1 nannan ⊢ φ ⊼ ψ ⊼ ψ ↔ φ → ψ ∧ ψ
2 anidmdbi ⊢ φ → ψ ∧ ψ ↔ φ → ψ
3 1 2 bitr2i ⊢ φ → ψ ↔ φ ⊼ ψ ⊼ ψ