Theorem ne3anior 2603

Description: A De Morgan's law for inequality. (Contributed by NM, 30-Sep-2013.)

Assertion

Ref	Expression
ne3anior	⊢ ((A ≠ B ∧ C ≠ D ∧ E ≠ F) ↔ ¬ (A = B ∨ C = D ∨ E = F))

Proof of Theorem ne3anior

Step	Hyp	Ref	Expression
1		3anor 948	. 2 ⊢ ((A ≠ B ∧ C ≠ D ∧ E ≠ F) ↔ ¬ (¬ A ≠ B ∨ ¬ C ≠ D ∨ ¬ E ≠ F))
2		nne 2521	. . 3 ⊢ (¬ A ≠ B ↔ A = B)
3		nne 2521	. . 3 ⊢ (¬ C ≠ D ↔ C = D)
4		nne 2521	. . 3 ⊢ (¬ E ≠ F ↔ E = F)
5	2, 3, 4	3orbi123i 1141	. 2 ⊢ ((¬ A ≠ B ∨ ¬ C ≠ D ∨ ¬ E ≠ F) ↔ (A = B ∨ C = D ∨ E = F))
6	1, 5	xchbinx 301	1 ⊢ ((A ≠ B ∧ C ≠ D ∧ E ≠ F) ↔ ¬ (A = B ∨ C = D ∨ E = F))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 176   ∨ w3o 933   ∧ w3a 934   = 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-3or 935  df-3an 936  df-ne 2519

This theorem is used by: (None)