Theorem ianordc 901

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, pm3.14 755, 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 𝜑 → (¬ (𝜑 ∧ 𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓)))

Proof of Theorem ianordc

Step	Hyp	Ref	Expression
1		imnan 692	. 2 ⊢ ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))
2		pm4.62dc 900	. 2 ⊢ (DECID 𝜑 → ((𝜑 → ¬ 𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓)))
3	1, 2	bitr3id 194	1 ⊢ (DECID 𝜑 → (¬ (𝜑 ∧ 𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 710  DECID wdc 836

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 711

This theorem depends on definitions:  df-bi 117  df-dc 837

This theorem is referenced by:  anordc  959  19.33bdc  1654  nn0n0n1ge2b  9472  nelfzo  10294  gcdsupex  12353  gcdsupcl  12354  dfgcd2  12410