Theorem pm2.18dc 750

Description: Proof by contradiction for a decidable proposition. Based on Theorem *2.18 of [WhiteheadRussell] p. 103 (also called the Law of Clavius). Intuitionistically it requires a decidability assumption, but compare with pm2.01 546 which does not. (Contributed by Jim Kingdon, 24-Mar-2018.)

Assertion

Ref	Expression
pm2.18dc	|- (DECID ph -> ( ( -. ph -> ph ) -> ph ) )

Proof of Theorem pm2.18dc

Step	Hyp	Ref	Expression
1		pm2.21 547	. . . 4 |- ( -. ph -> ( ph -> -. ( -. ph -> ph ) ) )
2	1	a2i 11	. . 3 |- ( ( -. ph -> ph ) -> ( -. ph -> -. ( -. ph -> ph ) ) )
3		condc 749	. . 3 |- (DECID ph -> ( ( -. ph -> -. ( -. ph -> ph ) ) -> ( ( -. ph -> ph ) -> ph ) ) )
4	2, 3	syl5 28	. 2 |- (DECID ph -> ( ( -. ph -> ph ) -> ( ( -. ph -> ph ) -> ph ) ) )
5	4	pm2.43d 44	1 |- (DECID ph -> ( ( -. ph -> ph ) -> ph ) )

Syntax hints:   -. wn 3   -> wi 4  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:  pm4.81dc  814