Theorem con4biddc 787

Description: A contraposition deduction. (Contributed by Jim Kingdon, 18-May-2018.)

Hypothesis

Ref	Expression
con4biddc.1	|- ( ph -> (DECID ps -> (DECID ch -> ( -. ps <-> -. ch ) ) ) )

Assertion

Ref	Expression
con4biddc	|- ( ph -> (DECID ps -> (DECID ch -> ( ps <-> ch ) ) ) )

Proof of Theorem con4biddc

Step	Hyp	Ref	Expression
1		con4biddc.1	. . . . . 6 |- ( ph -> (DECID ps -> (DECID ch -> ( -. ps <-> -. ch ) ) ) )
2		bi2 128	. . . . . 6 |- ( ( -. ps <-> -. ch ) -> ( -. ch -> -. ps ) )
3	1, 2	syl8 70	. . . . 5 |- ( ph -> (DECID ps -> (DECID ch -> ( -. ch -> -. ps ) ) ) )
4		condc 782	. . . . . 6 |- (DECID ch -> ( ( -. ch -> -. ps ) -> ( ps -> ch ) ) )
5	4	a2i 11	. . . . 5 |- ( (DECID ch -> ( -. ch -> -. ps ) ) -> (DECID ch -> ( ps -> ch ) ) )
6	3, 5	syl6 33	. . . 4 |- ( ph -> (DECID ps -> (DECID ch -> ( ps -> ch ) ) ) )
7	6	imp31 252	. . 3 |- ( ( ( ph /\ DECID ps ) /\ DECID ch ) -> ( ps -> ch ) )
8		bi1 116	. . . . . 6 |- ( ( -. ps <-> -. ch ) -> ( -. ps -> -. ch ) )
9	1, 8	syl8 70	. . . . 5 |- ( ph -> (DECID ps -> (DECID ch -> ( -. ps -> -. ch ) ) ) )
10		condc 782	. . . . . 6 |- (DECID ps -> ( ( -. ps -> -. ch ) -> ( ch -> ps ) ) )
11	10	imim2d 53	. . . . 5 |- (DECID ps -> ( (DECID ch -> ( -. ps -> -. ch ) ) -> (DECID ch -> ( ch -> ps ) ) ) )
12	9, 11	sylcom 28	. . . 4 |- ( ph -> (DECID ps -> (DECID ch -> ( ch -> ps ) ) ) )
13	12	imp31 252	. . 3 |- ( ( ( ph /\ DECID ps ) /\ DECID ch ) -> ( ch -> ps ) )
14	7, 13	impbid 127	. 2 |- ( ( ( ph /\ DECID ps ) /\ DECID ch ) -> ( ps <-> ch ) )
15	14	exp31 356	1 |- ( ph -> (DECID ps -> (DECID ch -> ( ps <-> ch ) ) ) )

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-in2 577  ax-io 662

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

This theorem is referenced by:  necon4abiddc  2318