Theorem con1biimdc 874

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		biimp 118	. . 3 |- ( ( -. ph <-> ps ) -> ( -. ph -> ps ) )
2		con1dc 857	. . 3 |- (DECID ph -> ( ( -. ph -> ps ) -> ( -. ps -> ph ) ) )
3	1, 2	syl5 32	. 2 |- (DECID ph -> ( ( -. ph <-> ps ) -> ( -. ps -> ph ) ) )
4		biimpr 130	. . . 4 |- ( ( -. ph <-> ps ) -> ( ps -> -. ph ) )
5	4	con2d 625	. . 3 |- ( ( -. ph <-> ps ) -> ( ph -> -. ps ) )
6	5	a1i 9	. 2 |- (DECID ph -> ( ( -. ph <-> ps ) -> ( ph -> -. ps ) ) )
7	3, 6	impbidd 127	1 |- (DECID ph -> ( ( -. ph <-> ps ) -> ( -. ps <-> ph ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105  DECID wdc 835

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836

This theorem is referenced by:  con1bidc  875  con1biddc  877