Theorem ianordc 889

Description: Negated conjunction in terms of disjunction (DeMorgan's law). Theorem *4.51 of [WhiteheadRussell] p. 120, but where one proposition is decidable. The reverse direction, pm3.14 743, holds for all propositions, but the equivalence only holds where one proposition is decidable. (Contributed by Jim Kingdon, 21-Apr-2018.)

Assertion

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

Proof of Theorem ianordc

Step	Hyp	Ref	Expression
1		imnan 680	. 2 |- ( ( ph -> -. ps ) <-> -. ( ph /\ ps ) )
2		pm4.62dc 888	. 2 |- (DECID ph -> ( ( ph -> -. ps ) <-> ( -. ph \/ -. ps ) ) )
3	1, 2	bitr3id 193	1 |- (DECID ph -> ( -. ( ph /\ ps ) <-> ( -. ph \/ -. ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698  DECID wdc 824

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 604  ax-in2 605  ax-io 699

This theorem depends on definitions:  df-bi 116  df-dc 825

This theorem is referenced by:  anordc  946  19.33bdc  1618  nn0n0n1ge2b  9270  gcdsupex  11890  gcdsupcl  11891  dfgcd2  11947