Theorem impidc 858

Description: An importation inference for a decidable consequent. (Contributed by Jim Kingdon, 30-Apr-2018.)

Hypothesis

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

Assertion

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

Proof of Theorem impidc

Step	Hyp	Ref	Expression
1		impidc.1	. . . . . 6 |- (DECID ch -> ( ph -> ( ps -> ch ) ) )
2	1	imp 124	. . . . 5 |- ( (DECID ch /\ ph ) -> ( ps -> ch ) )
3	2	con3d 631	. . . 4 |- ( (DECID ch /\ ph ) -> ( -. ch -> -. ps ) )
4	3	ex 115	. . 3 |- (DECID ch -> ( ph -> ( -. ch -> -. ps ) ) )
5	4	com23 78	. 2 |- (DECID ch -> ( -. ch -> ( ph -> -. ps ) ) )
6		con1dc 856	. 2 |- (DECID ch -> ( ( -. ch -> ( ph -> -. ps ) ) -> ( -. ( ph -> -. ps ) -> ch ) ) )
7	5, 6	mpd 13	1 |- (DECID ch -> ( -. ( ph -> -. ps ) -> ch ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104  DECID wdc 834

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 614  ax-in2 615  ax-io 709

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835

This theorem is referenced by:  simprimdc  859