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Theorem ianordc 798
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, pm3.14 669, 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
StepHypRef Expression
1 imnan 623 . 2 ((φ → ¬ ψ) ↔ ¬ (φ ψ))
2 pm4.62dc 797 . 2 (DECID φ → ((φ → ¬ ψ) ↔ (¬ φ ¬ ψ)))
31, 2syl5bbr 183 1 (DECID φ → (¬ (φ ψ) ↔ (¬ φ ¬ ψ)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   wo 628  DECID wdc 741
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 629
This theorem depends on definitions:  df-bi 110  df-dc 742
This theorem is referenced by:  anordc  862  19.33bdc  1518  nn0n0n1ge2b  8096
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