Theorem con1dc 846

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 619	. . 3 |- ( ps -> -. -. ps )
2	1	imim2i 12	. 2 |- ( ( -. ph -> ps ) -> ( -. ph -> -. -. ps ) )
3		condc 843	. 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 824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825

This theorem is referenced by:  impidc  848  simplimdc  850  con1biimdc  863  con1bdc  868  pm3.13dc  949  necon1aidc  2387  necon1bidc  2388  necon1addc  2412  necon1bddc  2413  phiprmpw  12154  fldivp1  12278  prmpwdvds  12285