Theorem pm2.18dc 856

Description: Proof by contradiction for a decidable proposition. Based on Theorem *2.18 of [WhiteheadRussell] p. 103 (also called Clavius law). Intuitionistically it requires a decidability assumption, but compare with pm2.01 617 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		pm2.21 618	. . . 4 |- ( -. ph -> ( ph -> -. ( -. ph -> ph ) ) )
2	1	a2i 11	. . 3 |- ( ( -. ph -> ph ) -> ( -. ph -> -. ( -. ph -> ph ) ) )
3		condc 854	. . 3 |- (DECID ph -> ( ( -. ph -> -. ( -. ph -> ph ) ) -> ( ( -. ph -> ph ) -> ph ) ) )
4	2, 3	syl5 32	. 2 |- (DECID ph -> ( ( -. ph -> ph ) -> ( ( -. ph -> ph ) -> ph ) ) )
5	4	pm2.43d 50	1 |- (DECID ph -> ( ( -. ph -> ph ) -> ph ) )

Syntax hints:   -. wn 3   -> wi 4  DECID wdc 835

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-in1 615  ax-in2 616  ax-io 710

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836

This theorem is referenced by:  pm4.81dc  909