Theorem con1dc 857

Description: Contraposition for a decidable proposition. Based on theorem *2.15 of [WhiteheadRussell] p. 102. (Contributed by Jim Kingdon, 29-Mar-2018.)

Assertion

Ref	Expression
con1dc	|- (DECID ph -> ( ( -. ph -> ps ) -> ( -. ps -> ph ) ) )

Proof of Theorem con1dc

Step	Hyp	Ref	Expression
1		notnot 630	. . 3 |- ( ps -> -. -. ps )
2	1	imim2i 12	. 2 |- ( ( -. ph -> ps ) -> ( -. ph -> -. -. ps ) )
3		condc 854	. 2 |- (DECID ph -> ( ( -. ph -> -. -. ps ) -> ( -. ps -> ph ) ) )
4	2, 3	syl5 32	1 |- (DECID ph -> ( ( -. ph -> ps ) -> ( -. ps -> 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:  impidc  859  simplimdc  861  con1biimdc  874  con1bdc  879  pm3.13dc  961  necon1aidc  2418  necon1bidc  2419  necon1addc  2443  necon1bddc  2444  exmidapne  7327  phiprmpw  12390  fldivp1  12517  prmpwdvds  12524