Theorem pm5.62dc 852

Description: Theorem *5.62 of [WhiteheadRussell] p. 125, for a decidable proposition. (Contributed by Jim Kingdon, 12-May-2018.)

Assertion

Ref	Expression
pm5.62dc	|- (DECID ps -> ( ( ( ph /\ ps ) \/ -. ps ) <-> ( ph \/ -. ps ) ) )

Proof of Theorem pm5.62dc

Step	Hyp	Ref	Expression
1		df-dc 743	. 2 |- (DECID ps <-> ( ps \/ -. ps ) )
2		ordir 730	. . . 4 |- ( ( ( ph /\ ps ) \/ -. ps ) <-> ( ( ph \/ -. ps ) /\ ( ps \/ -. ps ) ) )
3	2	simplbi 259	. . 3 |- ( ( ( ph /\ ps ) \/ -. ps ) -> ( ph \/ -. ps ) )
4	2	simplbi2 367	. . . 4 |- ( ( ph \/ -. ps ) -> ( ( ps \/ -. ps ) -> ( ( ph /\ ps ) \/ -. ps ) ) )
5	4	com12 27	. . 3 |- ( ( ps \/ -. ps ) -> ( ( ph \/ -. ps ) -> ( ( ph /\ ps ) \/ -. ps ) ) )
6	3, 5	impbid2 131	. 2 |- ( ( ps \/ -. ps ) -> ( ( ( ph /\ ps ) \/ -. ps ) <-> ( ph \/ -. ps ) ) )
7	1, 6	sylbi 114	1 |- (DECID ps -> ( ( ( ph /\ ps ) \/ -. ps ) <-> ( ph \/ -. ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   \/ wo 629  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-io 630

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

This theorem is referenced by: (None)