Theorem ianordc 911

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, pm3.14 765, 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 701	. 2 |- ( ( ph -> -. ps ) <-> -. ( ph /\ ps ) )
2		pm4.62dc 910	. 2 |- (DECID ph -> ( ( ph -> -. ps ) <-> ( -. ph \/ -. ps ) ) )
3	1, 2	bitr3id 194	1 |- (DECID ph -> ( -. ( ph /\ ps ) <-> ( -. ph \/ -. ps ) ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720  DECID wdc 846

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721

This proof depends on definitions:  df-bi 117  df-dc 847

This theorem is used by:  anordc  969  19.33bdc  1683  nn0n0n1ge2b  9725  nelfzo  10559  gcdsupex  12734  gcdsupcl  12735  dfgcd2  12791