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Theorem ianordc 835
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, pm3.14 703, 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 657 . 2 ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑𝜓))
2 pm4.62dc 834 . 2 (DECID 𝜑 → ((𝜑 → ¬ 𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓)))
31, 2syl5bbr 192 1 (DECID 𝜑 → (¬ (𝜑𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 662  DECID wdc 778
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663
This theorem depends on definitions:  df-bi 115  df-dc 779
This theorem is referenced by:  anordc  900  19.33bdc  1564  nn0n0n1ge2b  8736  gcdsupex  10743  gcdsupcl  10744  dfgcd2  10797
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