Theorem neanior 2602

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

Assertion

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

Proof of Theorem neanior

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	anbi12i 678	. 2 ⊢ ((A ≠ B ∧ C ≠ D) ↔ (¬ A = B ∧ ¬ C = D))
4		pm4.56 481	. 2 ⊢ ((¬ A = B ∧ ¬ C = D) ↔ ¬ (A = B ∨ C = D))
5	3, 4	bitri 240	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:  nelpri  3755