Theorem bijust 175

Description: Theorem used to justify definition of biconditional df-bi 177. (Contributed by NM, 11-May-1999.) (Proof shortened by Josh Purinton, 29-Dec-2000.)

Assertion

Ref	Expression
bijust	⊢ ¬ ((¬ ((φ → ψ) → ¬ (ψ → φ)) → ¬ ((φ → ψ) → ¬ (ψ → φ))) → ¬ (¬ ((φ → ψ) → ¬ (ψ → φ)) → ¬ ((φ → ψ) → ¬ (ψ → φ))))

Proof of Theorem bijust

Step	Hyp	Ref	Expression
1		id 19	. 2 ⊢ (¬ ((φ → ψ) → ¬ (ψ → φ)) → ¬ ((φ → ψ) → ¬ (ψ → φ)))
2		pm2.01 160	. 2 ⊢ (((¬ ((φ → ψ) → ¬ (ψ → φ)) → ¬ ((φ → ψ) → ¬ (ψ → φ))) → ¬ (¬ ((φ → ψ) → ¬ (ψ → φ)) → ¬ ((φ → ψ) → ¬ (ψ → φ)))) → ¬ (¬ ((φ → ψ) → ¬ (ψ → φ)) → ¬ ((φ → ψ) → ¬ (ψ → φ))))
3	1, 2	mt2 170	1 ⊢ ¬ ((¬ ((φ → ψ) → ¬ (ψ → φ)) → ¬ ((φ → ψ) → ¬ (ψ → φ))) → ¬ (¬ ((φ → ψ) → ¬ (ψ → φ)) → ¬ ((φ → ψ) → ¬ (ψ → φ))))

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is referenced by: (None)