Theorem con2biddc 870

Description: A contraposition deduction. (Contributed by Jim Kingdon, 11-Apr-2018.)

Hypothesis

Ref	Expression
con2biddc.1	|- ( ph -> (DECID ch -> ( ps <-> -. ch ) ) )

Assertion

Ref	Expression
con2biddc	|- ( ph -> (DECID ch -> ( ch <-> -. ps ) ) )

Proof of Theorem con2biddc

Step	Hyp	Ref	Expression
1		con2biddc.1	. . . 4 |- ( ph -> (DECID ch -> ( ps <-> -. ch ) ) )
2		bicom 139	. . . 4 |- ( ( ps <-> -. ch ) <-> ( -. ch <-> ps ) )
3	1, 2	syl6ib 160	. . 3 |- ( ph -> (DECID ch -> ( -. ch <-> ps ) ) )
4	3	con1biddc 866	. 2 |- ( ph -> (DECID ch -> ( -. ps <-> ch ) ) )
5		bicom 139	. 2 |- ( ( -. ps <-> ch ) <-> ( ch <-> -. ps ) )
6	4, 5	syl6ib 160	1 |- ( ph -> (DECID ch -> ( ch <-> -. ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> 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:  anordc  946  xor3dc  1377