Theorem neorian 2604

Description: A De Morgan's law for inequality. (Contributed by NM, 18-May-2007.)

Assertion

Ref	Expression
neorian	⊢ ((A ≠ B ∨ C ≠ D) ↔ ¬ (A = B ∧ C = D))

Proof of Theorem neorian

Step	Hyp	Ref	Expression
1		df-ne 2519	. . 3 ⊢ (A ≠ B ↔ ¬ A = B)
2		df-ne 2519	. . 3 ⊢ (C ≠ D ↔ ¬ C = D)
3	1, 2	orbi12i 507	. 2 ⊢ ((A ≠ B ∨ C ≠ D) ↔ (¬ A = B ∨ ¬ C = D))
4		ianor 474	. 2 ⊢ (¬ (A = B ∧ C = D) ↔ (¬ A = B ∨ ¬ C = D))
5	3, 4	bitr4i 243	1 ⊢ ((A ≠ B ∨ C ≠ D) ↔ ¬ (A = B ∧ C = D))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 176   ∨ wo 357   ∧ wa 358   = wceq 1642   ≠ wne 2517

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-ne 2519

This theorem is used by: (None)