Theorem peircedc 858

Description: Peirce's theorem for a decidable proposition. This odd-looking theorem can be seen as an alternative to exmiddc 782, condc 787, or notnotrdc 789 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 ph -> ( ( ( ph -> ps ) -> ph ) -> ph ) )

Proof of Theorem peircedc

Step	Hyp	Ref	Expression
1		df-dc 781	. 2 |- (DECID ph <-> ( ph \/ -. ph ) )
2		ax-1 5	. . 3 |- ( ph -> ( ( ( ph -> ps ) -> ph ) -> ph ) )
3		pm2.21 582	. . . . 5 |- ( -. ph -> ( ph -> ps ) )
4	3	imim1i 59	. . . 4 |- ( ( ( ph -> ps ) -> ph ) -> ( -. ph -> ph ) )
5	4	com12 30	. . 3 |- ( -. ph -> ( ( ( ph -> ps ) -> ph ) -> ph ) )
6	2, 5	jaoi 671	. 2 |- ( ( ph \/ -. ph ) -> ( ( ( ph -> ps ) -> ph ) -> ph ) )
7	1, 6	sylbi 119	1 |- (DECID ph -> ( ( ( ph -> ps ) -> ph ) -> ph ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 664  DECID wdc 780

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 580  ax-io 665

This theorem depends on definitions:  df-bi 115  df-dc 781

This theorem is referenced by:  looinvdc  859  exmoeudc  2011