Theorem pm5.6dc 868

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 869). (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		dfordc 824	. . 3 |- (DECID ps -> ( ( ps \/ ch ) <-> ( -. ps -> ch ) ) )
2	1	imbi2d 228	. 2 |- (DECID ps -> ( ( ph -> ( ps \/ ch ) ) <-> ( ph -> ( -. ps -> ch ) ) ) )
3		impexp 259	. 2 |- ( ( ( ph /\ -. ps ) -> ch ) <-> ( ph -> ( -. ps -> ch ) ) )
4	2, 3	syl6rbbr 197	1 |- (DECID ps -> ( ( ( ph /\ -. ps ) -> ch ) <-> ( ph -> ( ps \/ ch ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103   \/ wo 661  DECID wdc 775

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662

This theorem depends on definitions:  df-bi 115  df-dc 776

This theorem is referenced by: (None)