Metamath Proof Explorer

Theorem bj-stabpeirce

Description: Over minimal implicational calculus, Peirce's law is implied by the (classical refutation equivalent of) the double negation of the stability of any proposition. (Contributed by BJ, 30-Nov-2023) (Proof modification is discouraged.)

		Ref	Expression
	Assertion	bj-stabpeirce	$⊢ (((((φ \to ψ) \to ψ) \to φ) \to ψ) \to ψ) \to (((ψ \to φ) \to ψ) \to ψ)$

Proof

Step	Hyp	Ref	Expression
1		jarr	$⊢ (((φ \to ψ) \to ψ) \to φ) \to (ψ \to φ)$
2	1	imim1i	$⊢ ((ψ \to φ) \to ψ) \to ((((φ \to ψ) \to ψ) \to φ) \to ψ)$
3	2	imim1i	$⊢ (((((φ \to ψ) \to ψ) \to φ) \to ψ) \to ψ) \to (((ψ \to φ) \to ψ) \to ψ)$