Metamath Proof Explorer

Theorem re1luk2

Description: luk-2 derived from the Tarski-Bernays-Wajsberg axioms. (Contributed by Anthony Hart, 16-Aug-2011) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Assertion	re1luk2	$⊢ (\neg φ \to φ) \to φ$

Proof

Step	Hyp	Ref	Expression
1		tbw-negdf	$⊢ ((\neg φ \to (φ \to ⊥)) \to (((φ \to ⊥) \to \neg φ) \to ⊥)) \to ⊥$
2		tbw-ax2	$⊢ (((φ \to ⊥) \to \neg φ) \to ⊥) \to ((\neg φ \to (φ \to ⊥)) \to (((φ \to ⊥) \to \neg φ) \to ⊥))$
3		tbwlem4	$⊢ ((((φ \to ⊥) \to \neg φ) \to ⊥) \to ((\neg φ \to (φ \to ⊥)) \to (((φ \to ⊥) \to \neg φ) \to ⊥))) \to ((((\neg φ \to (φ \to ⊥)) \to (((φ \to ⊥) \to \neg φ) \to ⊥)) \to ⊥) \to ((φ \to ⊥) \to \neg φ))$
4	2 3	ax-mp	$⊢ (((\neg φ \to (φ \to ⊥)) \to (((φ \to ⊥) \to \neg φ) \to ⊥)) \to ⊥) \to ((φ \to ⊥) \to \neg φ)$
5	1 4	ax-mp	$⊢ (φ \to ⊥) \to \neg φ$
6		tbw-ax1	$⊢ ((φ \to ⊥) \to \neg φ) \to ((\neg φ \to φ) \to ((φ \to ⊥) \to φ))$
7	5 6	ax-mp	$⊢ (\neg φ \to φ) \to ((φ \to ⊥) \to φ)$
8		tbw-ax3	$⊢ ((φ \to ⊥) \to φ) \to φ$
9	7 8	tbwsyl	$⊢ (\neg φ \to φ) \to φ$