Metamath Proof Explorer

Theorem tbw-negdf

Description: The definition of negation, in terms of -> and F. . (Contributed by Anthony Hart, 15-Aug-2011) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Assertion	tbw-negdf	$⊢ ((\neg φ \to (φ \to ⊥)) \to (((φ \to ⊥) \to \neg φ) \to ⊥)) \to ⊥$

Proof

Step	Hyp	Ref	Expression
1		pm2.21	$⊢ \neg φ \to (φ \to ⊥)$
2		ax-1	$⊢ \neg φ \to ((φ \to ⊥) \to \neg φ)$
3		falim	$⊢ ⊥ \to ((φ \to ⊥) \to \neg φ)$
4	2 3	ja	$⊢ (φ \to ⊥) \to ((φ \to ⊥) \to \neg φ)$
5	4	pm2.43i	$⊢ (φ \to ⊥) \to \neg φ$
6	1 5	impbii	$⊢ \neg φ \leftrightarrow (φ \to ⊥)$
7		tbw-bijust	$⊢ (\neg φ \leftrightarrow (φ \to ⊥)) \leftrightarrow (((\neg φ \to (φ \to ⊥)) \to (((φ \to ⊥) \to \neg φ) \to ⊥)) \to ⊥)$
8	6 7	mpbi	$⊢ ((\neg φ \to (φ \to ⊥)) \to (((φ \to ⊥) \to \neg φ) \to ⊥)) \to ⊥$