Theorem dcfromcon 1469

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 855), 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 852	. . . . 5 ⊢ ¬ ¬ (𝜒 ∨ ¬ 𝜒)
2	1	pm2.21i 647	. . . 4 ⊢ (¬ (𝜒 ∨ ¬ 𝜒) → ¬ ⊤)
3		dcfromcon.3	. . . . 5 ⊢ ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑))
4		dcfromcon.1	. . . . . . 7 ⊢ (𝜑 ↔ (𝜒 ∨ ¬ 𝜒))
5	4	notbii 670	. . . . . 6 ⊢ (¬ 𝜑 ↔ ¬ (𝜒 ∨ ¬ 𝜒))
6		dcfromcon.2	. . . . . . 7 ⊢ (𝜓 ↔ ⊤)
7	6	notbii 670	. . . . . 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 1382	. 2 ⊢ (𝜒 ∨ ¬ 𝜒)
13		df-dc 837	. 2 ⊢ (DECID 𝜒 ↔ (𝜒 ∨ ¬ 𝜒))
14	12, 13	mpbir 146	1 ⊢ DECID 𝜒

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   ∨ wo 710  DECID wdc 836  ⊤wtru 1374

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

This theorem is referenced by: (None)