Theorem peircedc 914

Description: Peirce's theorem for a decidable proposition. This odd-looking theorem can be seen as an alternative to exmiddc 836, condc 853, or notnotrdc 843 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 835	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2		ax-1 6	. . 3 ⊢ (𝜑 → (((𝜑 → 𝜓) → 𝜑) → 𝜑))
3		pm2.21 617	. . . . 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 716	. 2 ⊢ ((𝜑 ∨ ¬ 𝜑) → (((𝜑 → 𝜓) → 𝜑) → 𝜑))
7	1, 6	sylbi 121	1 ⊢ (DECID 𝜑 → (((𝜑 → 𝜓) → 𝜑) → 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 708  DECID wdc 834

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-in2 615  ax-io 709

This theorem depends on definitions:  df-bi 117  df-dc 835

This theorem is referenced by:  looinvdc  915  exmoeudc  2089