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	⊢ ((𝜑 ∨ 𝜓) → ¬ (¬ 𝜑 ∧ ¬ 𝜓))

Proof of Theorem oranim

Step	Hyp	Ref	Expression
1		pm4.56 769	. . 3 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑 ∨ 𝜓))
2	1	biimpi 119	. 2 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ¬ (𝜑 ∨ 𝜓))
3	2	con2i 616	1 ⊢ ((𝜑 ∨ 𝜓) → ¬ (¬ 𝜑 ∧ ¬ 𝜓))

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  3310  prneimg  3696  ftpg  5597  xrlttri3  9576