Theorem ioran 476

Description: Negated disjunction in terms of conjunction (De Morgan's law). Compare Theorem *4.56 of [WhiteheadRussell] p. 120. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.)

Assertion

Ref	Expression
ioran	⊢ (¬ (φ ∨ ψ) ↔ (¬ φ ∧ ¬ ψ))

Proof of Theorem ioran

Step	Hyp	Ref	Expression
1		pm4.65 416	. 2 ⊢ (¬ (¬ φ → ψ) ↔ (¬ φ ∧ ¬ ψ))
2		pm4.64 361	. 2 ⊢ ((¬ φ → ψ) ↔ (φ ∨ ψ))
3	1, 2	xchnxbi 299	1 ⊢ (¬ (φ ∨ ψ) ↔ (¬ φ ∧ ¬ ψ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360

This theorem is referenced by:  pm4.56  481  oibabs  851  xor  861  3ioran  950  3ori  1242  ecase23d  1285  19.43OLD  1606  equvini  1987  2ralor  2781  dfun2  3491  sfin111  4537  nchoice  6309