Theorem pm5.6dc 916

Description: Conjunction in antecedent versus disjunction in consequent, for a decidable proposition. Theorem *5.6 of [WhiteheadRussell] p. 125, with decidability condition added. The reverse implication holds for all propositions (see pm5.6r 917). (Contributed by Jim Kingdon, 2-Apr-2018.)

Assertion

Ref	Expression
pm5.6dc	|- (DECID ps -> ( ( ( ph /\ -. ps ) -> ch ) <-> ( ph -> ( ps \/ ch ) ) ) )

Proof of Theorem pm5.6dc

Step	Hyp	Ref	Expression
1		impexp 261	. 2 |- ( ( ( ph /\ -. ps ) -> ch ) <-> ( ph -> ( -. ps -> ch ) ) )
2		dfordc 882	. . 3 |- (DECID ps -> ( ( ps \/ ch ) <-> ( -. ps -> ch ) ) )
3	2	imbi2d 229	. 2 |- (DECID ps -> ( ( ph -> ( ps \/ ch ) ) <-> ( ph -> ( -. ps -> ch ) ) ) )
4	1, 3	bitr4id 198	1 |- (DECID ps -> ( ( ( ph /\ -. ps ) -> ch ) <-> ( ph -> ( ps \/ ch ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698  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-dc 825

This theorem is referenced by: (None)