Theorem peircedc 900

Description: Peirce's theorem for a decidable proposition. This odd-looking theorem can be seen as an alternative to exmiddc 822, condc 839, or notnotrdc 829 in the sense of expressing the "difference" between an intuitionistic system of propositional calculus and a classical system. In intuitionistic logic, it only holds for decidable propositions. (Contributed by Jim Kingdon, 3-Jul-2018.)

Assertion

Ref	Expression
peircedc	⊢ (DECID 𝜑 → (((𝜑 → 𝜓) → 𝜑) → 𝜑))

Proof of Theorem peircedc

Step	Hyp	Ref	Expression
1		df-dc 821	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2		ax-1 6	. . 3 ⊢ (𝜑 → (((𝜑 → 𝜓) → 𝜑) → 𝜑))
3		pm2.21 607	. . . . 5 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
4	3	imim1i 60	. . . 4 ⊢ (((𝜑 → 𝜓) → 𝜑) → (¬ 𝜑 → 𝜑))
5	4	com12 30	. . 3 ⊢ (¬ 𝜑 → (((𝜑 → 𝜓) → 𝜑) → 𝜑))
6	2, 5	jaoi 706	. 2 ⊢ ((𝜑 ∨ ¬ 𝜑) → (((𝜑 → 𝜓) → 𝜑) → 𝜑))
7	1, 6	sylbi 120	1 ⊢ (DECID 𝜑 → (((𝜑 → 𝜓) → 𝜑) → 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 698  DECID wdc 820

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 605  ax-io 699

This theorem depends on definitions:  df-bi 116  df-dc 821

This theorem is referenced by:  looinvdc  901  exmoeudc  2063