Theorem peircedc 922

Description: Peirce's theorem for a decidable proposition. This odd-looking theorem can be seen as an alternative to exmiddc 844, condc 861, or notnotrdc 851 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 843	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2		ax-1 6	. . 3 ⊢ (𝜑 → (((𝜑 → 𝜓) → 𝜑) → 𝜑))
3		pm2.21 622	. . . . 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 724	. 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 716  DECID wdc 842

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 620  ax-io 717

This theorem depends on definitions:  df-bi 117  df-dc 843

This theorem is referenced by:  looinvdc  923  exmoeudc  2143  exmidpeirce  16709