Theorem pm3.13dc 900

Description: Theorem *3.13 of [WhiteheadRussell] p. 111, but for decidable propositions. The converse, pm3.14 702, holds for all propositions. (Contributed by Jim Kingdon, 22-Apr-2018.)

Assertion

Ref	Expression
pm3.13dc	|- (DECID ph -> (DECID ps -> ( -. ( ph /\ ps ) -> ( -. ph \/ -. ps ) ) ) )

Proof of Theorem pm3.13dc

Step	Hyp	Ref	Expression
1		dcn 779	. . 3 |- (DECID ph -> DECID -. ph )
2		dcn 779	. . 3 |- (DECID ps -> DECID -. ps )
3		dcor 876	. . 3 |- (DECID -. ph -> (DECID -. ps -> DECID ( -. ph \/ -. ps ) ) )
4	1, 2, 3	syl2im 38	. 2 |- (DECID ph -> (DECID ps -> DECID ( -. ph \/ -. ps ) ) )
5		pm3.11dc 898	. 2 |- (DECID ph -> (DECID ps -> ( -. ( -. ph \/ -. ps ) -> ( ph /\ ps ) ) ) )
6		con1dc 786	. 2 |- (DECID ( -. ph \/ -. ps ) -> ( ( -. ( -. ph \/ -. ps ) -> ( ph /\ ps ) ) -> ( -. ( ph /\ ps ) -> ( -. ph \/ -. ps ) ) ) )
7	4, 5, 6	syl6c 65	1 |- (DECID ph -> (DECID ps -> ( -. ( ph /\ ps ) -> ( -. ph \/ -. ps ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   \/ 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)