Theorem dcfrompeirce 1499

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 926), 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		pm2.67-2 725	. . . . 5 ⊢ (((𝜒 ∨ ¬ 𝜒) → ⊥) → (𝜒 → ⊥))
2		dfnot 1420	. . . . 5 ⊢ (¬ 𝜒 ↔ (𝜒 → ⊥))
3	1, 2	sylibr 134	. . . 4 ⊢ (((𝜒 ∨ ¬ 𝜒) → ⊥) → ¬ 𝜒)
4	3	olcd 746	. . 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 847	. 2 ⊢ (DECID 𝜒 ↔ (𝜒 ∨ ¬ 𝜒))
13	11, 12	mpbir 146	1 ⊢ DECID 𝜒

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   ∨ wo 720  DECID wdc 846  ⊥wfal 1407

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 623  ax-in2 624  ax-io 721

This theorem depends on definitions:  df-bi 117  df-dc 847  df-tru 1405  df-fal 1408

This theorem is referenced by: (None)