Theorem pm4.54dc 805

Description: Theorem *4.54 of [WhiteheadRussell] p. 120, for decidable propositions. One form of DeMorgan's law. (Contributed by Jim Kingdon, 2-May-2018.)

Assertion

Ref	Expression
pm4.54dc	|- (DECID ph -> (DECID ps -> ( ( -. ph /\ ps ) <-> -. ( ph \/ -. ps ) ) ) )

Proof of Theorem pm4.54dc

Step	Hyp	Ref	Expression
1		dcn 746	. . . . 5 |- (DECID ph -> DECID -. ph )
2		dfandc 778	. . . . 5 |- (DECID -. ph -> (DECID ps -> ( ( -. ph /\ ps ) <-> -. ( -. ph -> -. ps ) ) ) )
3	1, 2	syl 14	. . . 4 |- (DECID ph -> (DECID ps -> ( ( -. ph /\ ps ) <-> -. ( -. ph -> -. ps ) ) ) )
4	3	imp 115	. . 3 |- ( (DECID ph /\ DECID ps ) -> ( ( -. ph /\ ps ) <-> -. ( -. ph -> -. ps ) ) )
5		pm4.66dc 802	. . . . 5 |- (DECID ph -> ( ( -. ph -> -. ps ) <-> ( ph \/ -. ps ) ) )
6	5	adantr 261	. . . 4 |- ( (DECID ph /\ DECID ps ) -> ( ( -. ph -> -. ps ) <-> ( ph \/ -. ps ) ) )
7	6	notbid 592	. . 3 |- ( (DECID ph /\ DECID ps ) -> ( -. ( -. ph -> -. ps ) <-> -. ( ph \/ -. ps ) ) )
8	4, 7	bitrd 177	. 2 |- ( (DECID ph /\ DECID ps ) -> ( ( -. ph /\ ps ) <-> -. ( ph \/ -. ps ) ) )
9	8	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:  pm4.55dc  846