Theorem dcfromcon 1498

Description: The decidability of a proposition 𝜒 follows from a suitable instance of the principle of contraposition. Therefore, if we were to introduce contraposition as a general principle (without the decidability condition in condc 865), then we could prove that every proposition is decidable, giving us the classical system of propositional calculus (since the principle of contraposition is itself classically valid). (Contributed by Adrian Ducourtial, 6-Oct-2025.)

Hypotheses

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

Assertion

Ref	Expression
dcfromcon	⊢ DECID 𝜒

Proof of Theorem dcfromcon

Step	Hyp	Ref	Expression
1		nnexmid 862	. . . . 5 ⊢ ¬ ¬ (𝜒 ∨ ¬ 𝜒)
2	1	pm2.21i 655	. . . 4 ⊢ (¬ (𝜒 ∨ ¬ 𝜒) → ¬ ⊤)
3		dcfromcon.3	. . . . 5 ⊢ ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑))
4		dcfromcon.1	. . . . . . 7 ⊢ (𝜑 ↔ (𝜒 ∨ ¬ 𝜒))
5	4	notbii 678	. . . . . 6 ⊢ (¬ 𝜑 ↔ ¬ (𝜒 ∨ ¬ 𝜒))
6		dcfromcon.2	. . . . . . 7 ⊢ (𝜓 ↔ ⊤)
7	6	notbii 678	. . . . . 6 ⊢ (¬ 𝜓 ↔ ¬ ⊤)
8	5, 7	imbi12i 239	. . . . 5 ⊢ ((¬ 𝜑 → ¬ 𝜓) ↔ (¬ (𝜒 ∨ ¬ 𝜒) → ¬ ⊤))
9	6, 4	imbi12i 239	. . . . 5 ⊢ ((𝜓 → 𝜑) ↔ (⊤ → (𝜒 ∨ ¬ 𝜒)))
10	3, 8, 9	3imtr3i 200	. . . 4 ⊢ ((¬ (𝜒 ∨ ¬ 𝜒) → ¬ ⊤) → (⊤ → (𝜒 ∨ ¬ 𝜒)))
11	2, 10	ax-mp 5	. . 3 ⊢ (⊤ → (𝜒 ∨ ¬ 𝜒))
12	11	mptru 1411	. 2 ⊢ (𝜒 ∨ ¬ 𝜒)
13		df-dc 847	. 2 ⊢ (DECID 𝜒 ↔ (𝜒 ∨ ¬ 𝜒))
14	12, 13	mpbir 146	1 ⊢ DECID 𝜒

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   ∨ wo 720  DECID wdc 846  ⊤wtru 1403

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

This theorem is referenced by: (None)