Theorem con1biimdc 767

Description: Contraposition. (Contributed by Jim Kingdon, 4-Apr-2018.)

Assertion

Ref	Expression
con1biimdc	|- (DECID ph -> ( ( -. ph <-> ps ) -> ( -. ps <-> ph ) ) )

Proof of Theorem con1biimdc

Step	Hyp	Ref	Expression
1		bi1 111	. . 3 |- ( ( -. ph <-> ps ) -> ( -. ph -> ps ) )
2		con1dc 753	. . 3 |- (DECID ph -> ( ( -. ph -> ps ) -> ( -. ps -> ph ) ) )
3	1, 2	syl5 28	. 2 |- (DECID ph -> ( ( -. ph <-> ps ) -> ( -. ps -> ph ) ) )
4		bi2 121	. . . 4 |- ( ( -. ph <-> ps ) -> ( ps -> -. ph ) )
5	4	con2d 554	. . 3 |- ( ( -. ph <-> ps ) -> ( ph -> -. ps ) )
6	5	a1i 9	. 2 |- (DECID ph -> ( ( -. ph <-> ps ) -> ( ph -> -. ps ) ) )
7	3, 6	impbidd 118	1 |- (DECID ph -> ( ( -. ph <-> ps ) -> ( -. ps <-> ph ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 98  DECID wdc 742

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110  df-dc 743

This theorem is referenced by:  con1bidc  768  con1biddc  770