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Metamath Proof Explorer

Theorem tbwlem4

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	tbwlem4	$⊢ ((φ \to ⊥) \to ψ) \to ((ψ \to ⊥) \to φ)$

Proof

Step	Hyp	Ref	Expression
1		tbw-ax4	$⊢ ⊥ \to ⊥$
2		tbw-ax1	$⊢ (ψ \to ⊥) \to ((⊥ \to ⊥) \to (ψ \to ⊥))$
3		tbwlem1	$⊢ ((ψ \to ⊥) \to ((⊥ \to ⊥) \to (ψ \to ⊥))) \to ((⊥ \to ⊥) \to ((ψ \to ⊥) \to (ψ \to ⊥)))$
4	2 3	ax-mp	$⊢ (⊥ \to ⊥) \to ((ψ \to ⊥) \to (ψ \to ⊥))$
5	1 4	ax-mp	$⊢ (ψ \to ⊥) \to (ψ \to ⊥)$
6		tbwlem1	$⊢ ((ψ \to ⊥) \to (ψ \to ⊥)) \to (ψ \to ((ψ \to ⊥) \to ⊥))$
7	5 6	ax-mp	$⊢ ψ \to ((ψ \to ⊥) \to ⊥)$
8		tbw-ax1	$⊢ ((φ \to ⊥) \to ψ) \to ((ψ \to ((ψ \to ⊥) \to ⊥)) \to ((φ \to ⊥) \to ((ψ \to ⊥) \to ⊥)))$
9		tbwlem1	$⊢ (((φ \to ⊥) \to ψ) \to ((ψ \to ((ψ \to ⊥) \to ⊥)) \to ((φ \to ⊥) \to ((ψ \to ⊥) \to ⊥)))) \to ((ψ \to ((ψ \to ⊥) \to ⊥)) \to (((φ \to ⊥) \to ψ) \to ((φ \to ⊥) \to ((ψ \to ⊥) \to ⊥))))$
10	8 9	ax-mp	$⊢ (ψ \to ((ψ \to ⊥) \to ⊥)) \to (((φ \to ⊥) \to ψ) \to ((φ \to ⊥) \to ((ψ \to ⊥) \to ⊥)))$
11	7 10	ax-mp	$⊢ ((φ \to ⊥) \to ψ) \to ((φ \to ⊥) \to ((ψ \to ⊥) \to ⊥))$
12		tbwlem2	$⊢ ((φ \to ⊥) \to ((ψ \to ⊥) \to ⊥)) \to ((((φ \to ⊥) \to φ) \to φ) \to ((ψ \to ⊥) \to φ))$
13		tbwlem3	$⊢ ((((φ \to ⊥) \to φ) \to φ) \to ((ψ \to ⊥) \to φ)) \to ((ψ \to ⊥) \to φ)$
14	12 13	tbwsyl	$⊢ ((φ \to ⊥) \to ((ψ \to ⊥) \to ⊥)) \to ((ψ \to ⊥) \to φ)$
15	11 14	tbwsyl	$⊢ ((φ \to ⊥) \to ψ) \to ((ψ \to ⊥) \to φ)$