Theorem peircedc 921

Description: Peirce's theorem for a decidable proposition. This odd-looking theorem can be seen as an alternative to exmiddc 843, condc 860, or notnotrdc 850 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 842	. 2 |- (DECID ph <-> ( ph \/ -. ph ) )
2		ax-1 6	. . 3 |- ( ph -> ( ( ( ph -> ps ) -> ph ) -> ph ) )
3		pm2.21 622	. . . . 5 |- ( -. ph -> ( ph -> ps ) )
4	3	imim1i 60	. . . 4 |- ( ( ( ph -> ps ) -> ph ) -> ( -. ph -> ph ) )
5	4	com12 30	. . 3 |- ( -. ph -> ( ( ( ph -> ps ) -> ph ) -> ph ) )
6	2, 5	jaoi 723	. 2 |- ( ( ph \/ -. ph ) -> ( ( ( ph -> ps ) -> ph ) -> ph ) )
7	1, 6	sylbi 121	1 |- (DECID ph -> ( ( ( ph -> ps ) -> ph ) -> ph ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 715  DECID wdc 841

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 620  ax-io 716

This theorem depends on definitions:  df-bi 117  df-dc 842

This theorem is referenced by:  looinvdc  922  exmoeudc  2143