Theorem con2bidc 802

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 801	. . . . 5 |- (DECID ph -> (DECID ps -> ( ( -. ph <-> ps ) <-> ( -. ps <-> ph ) ) ) )
2	1	imp 122	. . . 4 |- ( (DECID ph /\ DECID ps ) -> ( ( -. ph <-> ps ) <-> ( -. ps <-> ph ) ) )
3		bicom 138	. . . 4 |- ( ( -. ph <-> ps ) <-> ( ps <-> -. ph ) )
4		bicom 138	. . . 4 |- ( ( -. ps <-> ph ) <-> ( ph <-> -. ps ) )
5	2, 3, 4	3bitr3g 220	. . 3 |- ( (DECID ph /\ DECID ps ) -> ( ( ps <-> -. ph ) <-> ( ph <-> -. ps ) ) )
6	5	bicomd 139	. 2 |- ( (DECID ph /\ DECID ps ) -> ( ( ph <-> -. ps ) <-> ( ps <-> -. ph ) ) )
7	6	ex 113	1 |- (DECID ph -> (DECID ps -> ( ( ph <-> -. ps ) <-> ( ps <-> -. ph ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103  DECID wdc 775

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662

This theorem depends on definitions:  df-bi 115  df-dc 776

This theorem is referenced by:  annimdc  878  pm4.55dc  879  orandc  880  nbbndc  1325