Theorem oranim 848

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	⊢ ((𝜑 ∨ 𝜓) → ¬ (¬ 𝜑 ∧ ¬ 𝜓))

Proof of Theorem oranim

Step	Hyp	Ref	Expression
1		pm4.56 847	. . 3 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑 ∨ 𝜓))
2	1	biimpi 119	. 2 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ¬ (𝜑 ∨ 𝜓))
3	2	con2i 595	1 ⊢ ((𝜑 ∨ 𝜓) → ¬ (¬ 𝜑 ∧ ¬ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 667

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  unssin  3254  prneimg  3640  ftpg  5520  xrlttri3  9366