Theorem pm3.11dc 864

Description: Theorem *3.11 of [WhiteheadRussell] p. 111, but for decidable propositions. The converse, pm3.1 671, 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 863	. . . 4 |- (DECID ph -> (DECID ps -> ( ( ph /\ ps ) <-> -. ( -. ph \/ -. ps ) ) ) )
2	1	imp 115	. . 3 |- ( (DECID ph /\ DECID ps ) -> ( ( ph /\ ps ) <-> -. ( -. ph \/ -. ps ) ) )
3	2	biimprd 147	. 2 |- ( (DECID ph /\ DECID ps ) -> ( -. ( -. ph \/ -. ps ) -> ( ph /\ ps ) ) )
4	3	ex 108	1 |- (DECID ph -> (DECID ps -> ( -. ( -. ph \/ -. ps ) -> ( ph /\ ps ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   \/ 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:  pm3.12dc  865  pm3.13dc  866