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Theorem ianordc 885
 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 𝜑 → (¬ (𝜑𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓)))

Proof of Theorem ianordc
StepHypRef Expression
1 imnan 680 . 2 ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑𝜓))
2 pm4.62dc 884 . 2 (DECID 𝜑 → ((𝜑 → ¬ 𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓)))
31, 2bitr3id 193 1 (DECID 𝜑 → (¬ (𝜑𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698  DECID wdc 820 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 821 This theorem is referenced by:  anordc  941  19.33bdc  1606  nn0n0n1ge2b  9183  gcdsupex  11718  gcdsupcl  11719  dfgcd2  11774
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