Theorem anordc 958

Description: Conjunction in terms of disjunction (DeMorgan's law). Theorem *4.5 of [WhiteheadRussell] p. 120, but where the propositions are decidable. The forward direction, pm3.1 755, holds for all propositions, but the equivalence only holds given decidability. (Contributed by Jim Kingdon, 21-Apr-2018.)

Assertion

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

Proof of Theorem anordc

Step	Hyp	Ref	Expression
1		dcan 935	. . 3 |- ( (DECID ph /\ DECID ps ) -> DECID ( ph /\ ps ) )
2	1	ex 115	. 2 |- (DECID ph -> (DECID ps -> DECID ( ph /\ ps ) ) )
3		ianordc 900	. . . . 5 |- (DECID ph -> ( -. ( ph /\ ps ) <-> ( -. ph \/ -. ps ) ) )
4	3, 3, 3	3bitr2rd 217	. . . 4 |- (DECID ph -> ( ( -. ph \/ -. ps ) <-> -. ( ph /\ ps ) ) )
5	4	a1d 22	. . 3 |- (DECID ph -> (DECID ( ph /\ ps ) -> ( ( -. ph \/ -. ps ) <-> -. ( ph /\ ps ) ) ) )
6	5	con2biddc 881	. 2 |- (DECID ph -> (DECID ( ph /\ ps ) -> ( ( ph /\ ps ) <-> -. ( -. ph \/ -. ps ) ) ) )
7	2, 6	syld 45	1 |- (DECID ph -> (DECID ps -> ( ( ph /\ ps ) <-> -. ( -. ph \/ -. ps ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709  DECID wdc 835

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836

This theorem is referenced by:  pm3.11dc  959  dn1dc  962