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Theorem oranim 770
 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 pm4.56 769 . . 3 ((¬ 𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑𝜓))
21biimpi 119 . 2 ((¬ 𝜑 ∧ ¬ 𝜓) → ¬ (𝜑𝜓))
32con2i 616 1 ((𝜑𝜓) → ¬ (¬ 𝜑 ∧ ¬ 𝜓))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 697 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 603  ax-in2 604  ax-io 698 This theorem depends on definitions:  df-bi 116 This theorem is referenced by:  unssin  3315  prneimg  3701  ftpg  5604  xrlttri3  9595
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