Theorem dcfromnotnotr 1458

Description: The decidability of a proposition 𝜓 follows from a suitable instance of double negation elimination (DNE). Therefore, if we were to introduce DNE as a general principle (without the decidability condition in notnotrdc 844), then we could prove that every proposition is decidable, giving us the classical system of propositional calculus (since DNE itself is classically valid). (Contributed by Adrian Ducourtial, 6-Oct-2025.)

Hypotheses

Ref	Expression
dcfromnotnotr.1	⊢ (𝜑 ↔ (𝜓 ∨ ¬ 𝜓))
dcfromnotnotr.2	⊢ (¬ ¬ 𝜑 → 𝜑)

Assertion

Ref	Expression
dcfromnotnotr	⊢ DECID 𝜓

Proof of Theorem dcfromnotnotr

Step	Hyp	Ref	Expression
1		nnexmid 851	. . 3 ⊢ ¬ ¬ (𝜓 ∨ ¬ 𝜓)
2		dcfromnotnotr.2	. . . 4 ⊢ (¬ ¬ 𝜑 → 𝜑)
3		dcfromnotnotr.1	. . . . . 6 ⊢ (𝜑 ↔ (𝜓 ∨ ¬ 𝜓))
4	3	notbii 669	. . . . 5 ⊢ (¬ 𝜑 ↔ ¬ (𝜓 ∨ ¬ 𝜓))
5	4	notbii 669	. . . 4 ⊢ (¬ ¬ 𝜑 ↔ ¬ ¬ (𝜓 ∨ ¬ 𝜓))
6	2, 5, 3	3imtr3i 200	. . 3 ⊢ (¬ ¬ (𝜓 ∨ ¬ 𝜓) → (𝜓 ∨ ¬ 𝜓))
7	1, 6	ax-mp 5	. 2 ⊢ (𝜓 ∨ ¬ 𝜓)
8		df-dc 836	. 2 ⊢ (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓))
9	7, 8	mpbir 146	1 ⊢ DECID 𝜓

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   ∨ wo 709  DECID wdc 835

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 710

This theorem depends on definitions:  df-bi 117  df-dc 836

This theorem is referenced by: (None)