Theorem orandc 934

Description: Disjunction in terms of conjunction (De Morgan's law), for decidable propositions. Compare Theorem *4.57 of [WhiteheadRussell] p. 120. (Contributed by Jim Kingdon, 13-Dec-2021.)

Assertion

Ref	Expression
orandc	|- ( (DECID ph /\ DECID ps ) -> ( ( ph \/ ps ) <-> -. ( -. ph /\ -. ps ) ) )

Proof of Theorem orandc

Step	Hyp	Ref	Expression
1		pm4.56 775	. 2 |- ( ( -. ph /\ -. ps ) <-> -. ( ph \/ ps ) )
2		dcn 837	. . . . 5 |- (DECID ph -> DECID -. ph )
3	2	adantr 274	. . . 4 |- ( (DECID ph /\ DECID ps ) -> DECID -. ph )
4		dcn 837	. . . . 5 |- (DECID ps -> DECID -. ps )
5	4	adantl 275	. . . 4 |- ( (DECID ph /\ DECID ps ) -> DECID -. ps )
6		dcan2 929	. . . 4 |- (DECID -. ph -> (DECID -. ps -> DECID ( -. ph /\ -. ps ) ) )
7	3, 5, 6	sylc 62	. . 3 |- ( (DECID ph /\ DECID ps ) -> DECID ( -. ph /\ -. ps ) )
8		dcor 930	. . . 4 |- (DECID ph -> (DECID ps -> DECID ( ph \/ ps ) ) )
9	8	imp 123	. . 3 |- ( (DECID ph /\ DECID ps ) -> DECID ( ph \/ ps ) )
10		con2bidc 870	. . 3 |- (DECID ( -. ph /\ -. ps ) -> (DECID ( ph \/ ps ) -> ( ( ( -. ph /\ -. ps ) <-> -. ( ph \/ ps ) ) <-> ( ( ph \/ ps ) <-> -. ( -. ph /\ -. ps ) ) ) ) )
11	7, 9, 10	sylc 62	. 2 |- ( (DECID ph /\ DECID ps ) -> ( ( ( -. ph /\ -. ps ) <-> -. ( ph \/ ps ) ) <-> ( ( ph \/ ps ) <-> -. ( -. ph /\ -. ps ) ) ) )
12	1, 11	mpbii 147	1 |- ( (DECID ph /\ DECID ps ) -> ( ( ph \/ ps ) <-> -. ( -. ph /\ -. ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 703  DECID wdc 829

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 609  ax-in2 610  ax-io 704

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830

This theorem is referenced by:  gcdaddm  11939