Theorem peircedc 813

Description: Peirce's theorem for a decidable proposition. This odd-looking theorem can be seen as an alternative to exmiddc 735, condc 740, or notnotdc 759 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 734	. 2 ⊢ (DECID φ ↔ (φ ∨ ¬ φ))
2		ax-1 5	. . 3 ⊢ (φ → (((φ → ψ) → φ) → φ))
3		pm2.21 535	. . . . 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 623	. 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 616  DECID wdc 733

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 533  ax-io 617

This theorem depends on definitions:  df-bi 110  df-dc 734

This theorem is referenced by:  looinvdc  814  exmoeudc  1945