Theorem anordc 897

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, pm3.1 703, 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 875	. 2 |- (DECID ph -> (DECID ps -> DECID ( ph /\ ps ) ) )
2		ianordc 832	. . . . 5 |- (DECID ph -> ( -. ( ph /\ ps ) <-> ( -. ph \/ -. ps ) ) )
3	2	bicomd 139	. . . 4 |- (DECID ph -> ( ( -. ph \/ -. ps ) <-> -. ( ph /\ ps ) ) )
4	3	a1d 22	. . 3 |- (DECID ph -> (DECID ( ph /\ ps ) -> ( ( -. ph \/ -. ps ) <-> -. ( ph /\ ps ) ) ) )
5	4	con2biddc 807	. 2 |- (DECID ph -> (DECID ( ph /\ ps ) -> ( ( ph /\ ps ) <-> -. ( -. ph \/ -. ps ) ) ) )
6	1, 5	syld 44	1 |- (DECID ph -> (DECID ps -> ( ( ph /\ ps ) <-> -. ( -. ph \/ -. ps ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103   \/ wo 661  DECID wdc 775

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662

This theorem depends on definitions:  df-bi 115  df-dc 776

This theorem is referenced by:  pm3.11dc  898  dn1dc  901