tbwlem5

Description: Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'. (Contributed by Anthony Hart, 16-Aug-2011) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Assertion	tbwlem5	$⊢ ((φ \to (ψ \to ⊥)) \to ⊥) \to φ$

Proof

Step	Hyp	Ref	Expression
1		tbw-ax2	$⊢ φ \to (ψ \to φ)$
2		tbw-ax1	$⊢ (ψ \to φ) \to ((φ \to ⊥) \to (ψ \to ⊥))$
3	1 2	tbwsyl	$⊢ φ \to ((φ \to ⊥) \to (ψ \to ⊥))$
4		tbwlem1	$⊢ (φ \to ((φ \to ⊥) \to (ψ \to ⊥))) \to ((φ \to ⊥) \to (φ \to (ψ \to ⊥)))$
5	3 4	ax-mp	$⊢ (φ \to ⊥) \to (φ \to (ψ \to ⊥))$
6		tbwlem4	$⊢ ((φ \to ⊥) \to (φ \to (ψ \to ⊥))) \to (((φ \to (ψ \to ⊥)) \to ⊥) \to φ)$
7	5 6	ax-mp	$⊢ ((φ \to (ψ \to ⊥)) \to ⊥) \to φ$

Metamath Proof Explorer

Theorem tbwlem5

Proof