Theorem pm4.63dc 876

Description: Theorem *4.63 of [WhiteheadRussell] p. 120, for decidable propositions. (Contributed by Jim Kingdon, 1-May-2018.)

Assertion

Ref	Expression
pm4.63dc	|- (DECID ph -> (DECID ps -> ( -. ( ph -> -. ps ) <-> ( ph /\ ps ) ) ) )

Proof of Theorem pm4.63dc

Step	Hyp	Ref	Expression
1		dfandc 874	. . . 4 |- (DECID ph -> (DECID ps -> ( ( ph /\ ps ) <-> -. ( ph -> -. ps ) ) ) )
2	1	imp 123	. . 3 |- ( (DECID ph /\ DECID ps ) -> ( ( ph /\ ps ) <-> -. ( ph -> -. ps ) ) )
3	2	bicomd 140	. 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  DECID wdc 824

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 604  ax-in2 605  ax-io 699

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825

This theorem is referenced by:  pm4.67dc  877