Theorem con2bidc 865

Description: Contraposition. (Contributed by Jim Kingdon, 17-Apr-2018.)

Assertion

Ref	Expression
con2bidc	|- (DECID ph -> (DECID ps -> ( ( ph <-> -. ps ) <-> ( ps <-> -. ph ) ) ) )

Proof of Theorem con2bidc

Step	Hyp	Ref	Expression
1		con1bidc 864	. . . . 5 |- (DECID ph -> (DECID ps -> ( ( -. ph <-> ps ) <-> ( -. ps <-> ph ) ) ) )
2	1	imp 123	. . . 4 |- ( (DECID ph /\ DECID ps ) -> ( ( -. ph <-> ps ) <-> ( -. ps <-> ph ) ) )
3		bicom 139	. . . 4 |- ( ( -. ph <-> ps ) <-> ( ps <-> -. ph ) )
4		bicom 139	. . . 4 |- ( ( -. ps <-> ph ) <-> ( ph <-> -. ps ) )
5	2, 3, 4	3bitr3g 221	. . 3 |- ( (DECID ph /\ DECID ps ) -> ( ( ps <-> -. ph ) <-> ( ph <-> -. ps ) ) )
6	5	bicomd 140	. 2 |- ( (DECID ph /\ DECID ps ) -> ( ( ph <-> -. ps ) <-> ( ps <-> -. ph ) ) )
7	6	ex 114	1 |- (DECID ph -> (DECID ps -> ( ( ph <-> -. ps ) <-> ( ps <-> -. ph ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104  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:  annimdc  927  pm4.55dc  928  orandc  929  nbbndc  1384