Theorem dcfrompeirce 1470

Description: The decidability of a proposition 𝜒 follows from a suitable instance of Peirce's law. Therefore, if we were to introduce Peirce's law as a general principle (without the decidability condition in peircedc 916), then we could prove that every proposition is decidable, giving us the classical system of propositional calculus (since Perice's law is itself classically valid). (Contributed by Adrian Ducourtial, 6-Oct-2025.)

Hypotheses

Ref	Expression
dcfrompeirce.1	⊢ (𝜑 ↔ (𝜒 ∨ ¬ 𝜒))
dcfrompeirce.2	⊢ (𝜓 ↔ ⊥)
dcfrompeirce.3	⊢ (((𝜑 → 𝜓) → 𝜑) → 𝜑)

Assertion

Ref	Expression
dcfrompeirce	⊢ DECID 𝜒

Proof of Theorem dcfrompeirce

Step	Hyp	Ref	Expression
1		pm2.67-2 715	. . . . 5 ⊢ (((𝜒 ∨ ¬ 𝜒) → ⊥) → (𝜒 → ⊥))
2		dfnot 1391	. . . . 5 ⊢ (¬ 𝜒 ↔ (𝜒 → ⊥))
3	1, 2	sylibr 134	. . . 4 ⊢ (((𝜒 ∨ ¬ 𝜒) → ⊥) → ¬ 𝜒)
4	3	olcd 736	. . 3 ⊢ (((𝜒 ∨ ¬ 𝜒) → ⊥) → (𝜒 ∨ ¬ 𝜒))
5		dcfrompeirce.3	. . . 4 ⊢ (((𝜑 → 𝜓) → 𝜑) → 𝜑)
6		dcfrompeirce.1	. . . . . 6 ⊢ (𝜑 ↔ (𝜒 ∨ ¬ 𝜒))
7		dcfrompeirce.2	. . . . . 6 ⊢ (𝜓 ↔ ⊥)
8	6, 7	imbi12i 239	. . . . 5 ⊢ ((𝜑 → 𝜓) ↔ ((𝜒 ∨ ¬ 𝜒) → ⊥))
9	8, 6	imbi12i 239	. . . 4 ⊢ (((𝜑 → 𝜓) → 𝜑) ↔ (((𝜒 ∨ ¬ 𝜒) → ⊥) → (𝜒 ∨ ¬ 𝜒)))
10	5, 9, 6	3imtr3i 200	. . 3 ⊢ ((((𝜒 ∨ ¬ 𝜒) → ⊥) → (𝜒 ∨ ¬ 𝜒)) → (𝜒 ∨ ¬ 𝜒))
11	4, 10	ax-mp 5	. 2 ⊢ (𝜒 ∨ ¬ 𝜒)
12		df-dc 837	. 2 ⊢ (DECID 𝜒 ↔ (𝜒 ∨ ¬ 𝜒))
13	11, 12	mpbir 146	1 ⊢ DECID 𝜒

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   ∨ wo 710  DECID wdc 836  ⊥wfal 1378

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711

This theorem depends on definitions:  df-bi 117  df-dc 837  df-tru 1376  df-fal 1379

This theorem is referenced by: (None)