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Theorem oranim 818
 Description: Disjunction in terms of conjunction (DeMorgan's law). One direction of Theorem *4.57 of [WhiteheadRussell] p. 120. The converse does not hold intuitionistically but does hold in classical logic. (Contributed by Jim Kingdon, 25-Jul-2018.)
Assertion
Ref Expression
oranim ((𝜑𝜓) → ¬ (¬ 𝜑 ∧ ¬ 𝜓))

Proof of Theorem oranim
StepHypRef Expression
1 pm4.56 817 . . 3 ((¬ 𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑𝜓))
21biimpi 117 . 2 ((¬ 𝜑 ∧ ¬ 𝜓) → ¬ (𝜑𝜓))
32con2i 567 1 ((𝜑𝜓) → ¬ (¬ 𝜑 ∧ ¬ 𝜓))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 101   ∨ wo 639 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640 This theorem depends on definitions:  df-bi 114 This theorem is referenced by:  unssin  3204  prneimg  3573  ftpg  5375  xrlttri3  8819
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