Theorem pm3.13dc 866

Description: Theorem *3.13 of [WhiteheadRussell] p. 111, but for decidable propositions. The converse, pm3.14 670, 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 746	. . 3 |- (DECID ph -> DECID -. ph )
2		dcn 746	. . 3 |- (DECID ps -> DECID -. ps )
3		dcor 843	. . 3 |- (DECID -. ph -> (DECID -. ps -> DECID ( -. ph \/ -. ps ) ) )
4	1, 2, 3	syl2im 34	. 2 |- (DECID ph -> (DECID ps -> DECID ( -. ph \/ -. ps ) ) )
5		pm3.11dc 864	. 2 |- (DECID ph -> (DECID ps -> ( -. ( -. ph \/ -. ps ) -> ( ph /\ ps ) ) ) )
6		con1dc 753	. 2 |- (DECID ( -. ph \/ -. ps ) -> ( ( -. ( -. ph \/ -. ps ) -> ( ph /\ ps ) ) -> ( -. ( ph /\ ps ) -> ( -. ph \/ -. ps ) ) ) )
7	4, 5, 6	syl6c 60	1 |- (DECID ph -> (DECID ps -> ( -. ( ph /\ ps ) -> ( -. ph \/ -. ps ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   \/ 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-in1 544  ax-in2 545  ax-io 630

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

This theorem is referenced by: (None)