Metamath Proof Explorer

Theorem tbwlem2

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	tbwlem2	$⊢ (φ \to (ψ \to ⊥)) \to (((φ \to χ) \to θ) \to (ψ \to θ))$

Proof

Step	Hyp	Ref	Expression
1		tbw-ax1	$⊢ (φ \to (ψ \to ⊥)) \to (((ψ \to ⊥) \to χ) \to (φ \to χ))$
2		tbw-ax4	$⊢ ⊥ \to χ$
3		tbw-ax1	$⊢ (ψ \to ⊥) \to ((⊥ \to χ) \to (ψ \to χ))$
4		tbwlem1	$⊢ ((ψ \to ⊥) \to ((⊥ \to χ) \to (ψ \to χ))) \to ((⊥ \to χ) \to ((ψ \to ⊥) \to (ψ \to χ)))$
5	3 4	ax-mp	$⊢ (⊥ \to χ) \to ((ψ \to ⊥) \to (ψ \to χ))$
6	2 5	ax-mp	$⊢ (ψ \to ⊥) \to (ψ \to χ)$
7		tbwlem1	$⊢ ((ψ \to ⊥) \to (ψ \to χ)) \to (ψ \to ((ψ \to ⊥) \to χ))$
8	6 7	ax-mp	$⊢ ψ \to ((ψ \to ⊥) \to χ)$
9		tbw-ax1	$⊢ (ψ \to ((ψ \to ⊥) \to χ)) \to ((((ψ \to ⊥) \to χ) \to (φ \to χ)) \to (ψ \to (φ \to χ)))$
10	8 9	ax-mp	$⊢ (((ψ \to ⊥) \to χ) \to (φ \to χ)) \to (ψ \to (φ \to χ))$
11	1 10	tbwsyl	$⊢ (φ \to (ψ \to ⊥)) \to (ψ \to (φ \to χ))$
12		tbw-ax1	$⊢ (ψ \to (φ \to χ)) \to (((φ \to χ) \to θ) \to (ψ \to θ))$
13	11 12	tbwsyl	$⊢ (φ \to (ψ \to ⊥)) \to (((φ \to χ) \to θ) \to (ψ \to θ))$