Theorem bj-peircestab 33888

Description: Over minimal implicational calculus, Peirce's law implies the double negation of the stability of any proposition (that is the interpretation when ⊥ is substitued for 𝜓). (Contributed by BJ, 30-Nov-2023.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-peircestab	⊢ (((((𝜑 → 𝜓) → 𝜓) → 𝜑) → 𝜓) → 𝜓)

Proof of Theorem bj-peircestab

Step	Hyp	Ref	Expression
1		id 22	. . . 4 ⊢ (((((𝜑 → 𝜓) → 𝜓) → 𝜑) → 𝜓) → ((((𝜑 → 𝜓) → 𝜓) → 𝜑) → 𝜓))
2	1	a2i 14	. . 3 ⊢ ((((((𝜑 → 𝜓) → 𝜓) → 𝜑) → 𝜓) → (((𝜑 → 𝜓) → 𝜓) → 𝜑)) → (((((𝜑 → 𝜓) → 𝜓) → 𝜑) → 𝜓) → 𝜓))
3		ax-1 6	. . . . . . 7 ⊢ (𝜓 → ((𝜑 → 𝜓) → 𝜓))
4	3	imim2i 16	. . . . . 6 ⊢ (((((𝜑 → 𝜓) → 𝜓) → 𝜑) → 𝜓) → ((((𝜑 → 𝜓) → 𝜓) → 𝜑) → ((𝜑 → 𝜓) → 𝜓)))
5		peirce 204	. . . . . 6 ⊢ (((((𝜑 → 𝜓) → 𝜓) → 𝜑) → ((𝜑 → 𝜓) → 𝜓)) → ((𝜑 → 𝜓) → 𝜓))
6	4, 5	syl 17	. . . . 5 ⊢ (((((𝜑 → 𝜓) → 𝜓) → 𝜑) → 𝜓) → ((𝜑 → 𝜓) → 𝜓))
7		imim2 58	. . . . 5 ⊢ ((𝜓 → 𝜑) → (((𝜑 → 𝜓) → 𝜓) → ((𝜑 → 𝜓) → 𝜑)))
8		peirce 204	. . . . 5 ⊢ (((𝜑 → 𝜓) → 𝜑) → 𝜑)
9	6, 7, 8	syl56 36	. . . 4 ⊢ ((𝜓 → 𝜑) → (((((𝜑 → 𝜓) → 𝜓) → 𝜑) → 𝜓) → 𝜑))
10	9	a1dd 50	. . 3 ⊢ ((𝜓 → 𝜑) → (((((𝜑 → 𝜓) → 𝜓) → 𝜑) → 𝜓) → (((𝜑 → 𝜓) → 𝜓) → 𝜑)))
11	2, 10	syl11 33	. 2 ⊢ (((((𝜑 → 𝜓) → 𝜓) → 𝜑) → 𝜓) → ((𝜓 → 𝜑) → 𝜓))
12		peirce 204	. 2 ⊢ (((𝜓 → 𝜑) → 𝜓) → 𝜓)
13	11, 12	syl 17	1 ⊢ (((((𝜑 → 𝜓) → 𝜓) → 𝜑) → 𝜓) → 𝜓)

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is referenced by: (None)