Theorem peircedc 820

Description: Peirce's theorem for a decidable proposition. This odd-looking theorem can be seen as an alternative to exmiddc 744, condc 749, or notnotrdc 751 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 743	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2		ax-1 5	. . 3 ⊢ (𝜑 → (((𝜑 → 𝜓) → 𝜑) → 𝜑))
3		pm2.21 547	. . . . 5 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
4	3	imim1i 54	. . . 4 ⊢ (((𝜑 → 𝜓) → 𝜑) → (¬ 𝜑 → 𝜑))
5	4	com12 27	. . 3 ⊢ (¬ 𝜑 → (((𝜑 → 𝜓) → 𝜑) → 𝜑))
6	2, 5	jaoi 636	. 2 ⊢ ((𝜑 ∨ ¬ 𝜑) → (((𝜑 → 𝜓) → 𝜑) → 𝜑))
7	1, 6	sylbi 114	1 ⊢ (DECID 𝜑 → (((𝜑 → 𝜓) → 𝜑) → 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 629  DECID wdc 742

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110  df-dc 743

This theorem is referenced by:  looinvdc  821  exmoeudc  1963