Theorem ianordc 884

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, pm3.14 742, 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 679	. 2 |- ( ( ph -> -. ps ) <-> -. ( ph /\ ps ) )
2		pm4.62dc 883	. 2 |- (DECID ph -> ( ( ph -> -. ps ) <-> ( -. ph \/ -. ps ) ) )
3	1, 2	syl5bbr 193	1 |- (DECID ph -> ( -. ( ph /\ ps ) <-> ( -. ph \/ -. ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 697  DECID wdc 819

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 603  ax-in2 604  ax-io 698

This theorem depends on definitions:  df-bi 116  df-dc 820

This theorem is referenced by:  anordc  940  19.33bdc  1609  nn0n0n1ge2b  9123  gcdsupex  11635  gcdsupcl  11636  dfgcd2  11691