Theorem pm4.63dc 780

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 778	. . . 4 |- (DECID ph -> (DECID ps -> ( ( ph /\ ps ) <-> -. ( ph -> -. ps ) ) ) )
2	1	imp 115	. . 3 |- ( (DECID ph /\ DECID ps ) -> ( ( ph /\ ps ) <-> -. ( ph -> -. ps ) ) )
3	2	bicomd 129	. 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  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:  pm4.67dc  781