Theorem peircedc 853

Description: Peirce's theorem for a decidable proposition. This odd-looking theorem can be seen as an alternative to exmiddc 777, condc 782, or notnotrdc 784 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 776	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2		ax-1 5	. . 3 ⊢ (𝜑 → (((𝜑 → 𝜓) → 𝜑) → 𝜑))
3		pm2.21 579	. . . . 5 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
4	3	imim1i 59	. . . 4 ⊢ (((𝜑 → 𝜓) → 𝜑) → (¬ 𝜑 → 𝜑))
5	4	com12 30	. . 3 ⊢ (¬ 𝜑 → (((𝜑 → 𝜓) → 𝜑) → 𝜑))
6	2, 5	jaoi 668	. 2 ⊢ ((𝜑 ∨ ¬ 𝜑) → (((𝜑 → 𝜓) → 𝜑) → 𝜑))
7	1, 6	sylbi 119	1 ⊢ (DECID 𝜑 → (((𝜑 → 𝜓) → 𝜑) → 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 661  DECID wdc 775

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 577  ax-io 662

This theorem depends on definitions:  df-bi 115  df-dc 776

This theorem is referenced by:  looinvdc  854  exmoeudc  2004