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Theorem ianordc 799
 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, pm3.14 670, 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 624 . 2 ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑𝜓))
2 pm4.62dc 798 . 2 (DECID 𝜑 → ((𝜑 → ¬ 𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓)))
31, 2syl5bbr 183 1 (DECID 𝜑 → (¬ (𝜑𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629  DECID wdc 742 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630 This theorem depends on definitions:  df-bi 110  df-dc 743 This theorem is referenced by:  anordc  863  19.33bdc  1521  nn0n0n1ge2b  8320
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