Metamath Proof Explorer

Theorem tbwlem1

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	tbwlem1	$⊢ (φ \to (ψ \to χ)) \to (ψ \to (φ \to χ))$

Proof

Step	Hyp	Ref	Expression
1		tbw-ax1	$⊢ (φ \to (ψ \to χ)) \to (((ψ \to χ) \to χ) \to (φ \to χ))$
2		tbw-ax2	$⊢ ψ \to ((ψ \to χ) \to ψ)$
3		tbw-ax1	$⊢ ((ψ \to χ) \to ψ) \to ((ψ \to χ) \to ((ψ \to χ) \to χ))$
4	2 3	tbwsyl	$⊢ ψ \to ((ψ \to χ) \to ((ψ \to χ) \to χ))$
5		tbw-ax1	$⊢ ((ψ \to χ) \to ((ψ \to χ) \to χ)) \to ((((ψ \to χ) \to χ) \to χ) \to ((ψ \to χ) \to χ))$
6		tbw-ax3	$⊢ ((((ψ \to χ) \to χ) \to χ) \to ((ψ \to χ) \to χ)) \to ((ψ \to χ) \to χ)$
7	5 6	tbwsyl	$⊢ ((ψ \to χ) \to ((ψ \to χ) \to χ)) \to ((ψ \to χ) \to χ)$
8	4 7	tbwsyl	$⊢ ψ \to ((ψ \to χ) \to χ)$
9		tbw-ax1	$⊢ (ψ \to ((ψ \to χ) \to χ)) \to ((((ψ \to χ) \to χ) \to (φ \to χ)) \to (ψ \to (φ \to χ)))$
10	8 9	ax-mp	$⊢ (((ψ \to χ) \to χ) \to (φ \to χ)) \to (ψ \to (φ \to χ))$
11	1 10	tbwsyl	$⊢ (φ \to (ψ \to χ)) \to (ψ \to (φ \to χ))$