Theorem pm3.11dc 941

Description: Theorem *3.11 of [WhiteheadRussell] p. 111, but for decidable propositions. The converse, pm3.1 743, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 22-Apr-2018.)

Assertion

Ref	Expression
pm3.11dc	|- (DECID ph -> (DECID ps -> ( -. ( -. ph \/ -. ps ) -> ( ph /\ ps ) ) ) )

Proof of Theorem pm3.11dc

Step	Hyp	Ref	Expression
1		anordc 940	. . . 4 |- (DECID ph -> (DECID ps -> ( ( ph /\ ps ) <-> -. ( -. ph \/ -. ps ) ) ) )
2	1	imp 123	. . 3 |- ( (DECID ph /\ DECID ps ) -> ( ( ph /\ ps ) <-> -. ( -. ph \/ -. ps ) ) )
3	2	biimprd 157	. 2 |- ( (DECID ph /\ DECID ps ) -> ( -. ( -. ph \/ -. ps ) -> ( ph /\ ps ) ) )
4	3	ex 114	1 |- (DECID ph -> (DECID ps -> ( -. ( -. ph \/ -. ps ) -> ( ph /\ ps ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 697  DECID wdc 819

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 603  ax-in2 604  ax-io 698

This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820

This theorem is referenced by:  pm3.12dc  942  pm3.13dc  943