Theorem con4biddc 847

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		biimpr 129	. . . . . 6 |- ( ( -. ps <-> -. ch ) -> ( -. ch -> -. ps ) )
3	1, 2	syl8 71	. . . . 5 |- ( ph -> (DECID ps -> (DECID ch -> ( -. ch -> -. ps ) ) ) )
4		condc 843	. . . . . 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 254	. . 3 |- ( ( ( ph /\ DECID ps ) /\ DECID ch ) -> ( ps -> ch ) )
8		biimp 117	. . . . . 6 |- ( ( -. ps <-> -. ch ) -> ( -. ps -> -. ch ) )
9	1, 8	syl8 71	. . . . 5 |- ( ph -> (DECID ps -> (DECID ch -> ( -. ps -> -. ch ) ) ) )
10		condc 843	. . . . . 6 |- (DECID ps -> ( ( -. ps -> -. ch ) -> ( ch -> ps ) ) )
11	10	imim2d 54	. . . . 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 254	. . 3 |- ( ( ( ph /\ DECID ps ) /\ DECID ch ) -> ( ch -> ps ) )
14	7, 13	impbid 128	. 2 |- ( ( ( ph /\ DECID ps ) /\ DECID ch ) -> ( ps <-> ch ) )
15	14	exp31 362	1 |- ( ph -> (DECID ps -> (DECID ch -> ( ps <-> ch ) ) ) )

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:  necon4abiddc  2409