Theorem peircedc 819

Description: Peirce's theorem for a decidable proposition. This odd-looking theorem can be seen as an alternative to exmiddc 743, condc 748, or notnotdc 765 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 742	. 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 635	. 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 628  DECID wdc 741

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 629

This theorem depends on definitions:  df-bi 110  df-dc 742

This theorem is referenced by:  looinvdc  820  exmoeudc  1960