Theorem neanior 2444

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 2358	. . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
2		df-ne 2358	. . 3 ⊢ (𝐶 ≠ 𝐷 ↔ ¬ 𝐶 = 𝐷)
3	1, 2	anbi12i 460	. 2 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐷) ↔ (¬ 𝐴 = 𝐵 ∧ ¬ 𝐶 = 𝐷))
4		pm4.56 781	. 2 ⊢ ((¬ 𝐴 = 𝐵 ∧ ¬ 𝐶 = 𝐷) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐶 = 𝐷))
5	3, 4	bitri 184	1 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐷) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐶 = 𝐷))

Syntax hints:  ¬ wn 3   ∧ wa 104   ↔ wb 105   ∨ wo 709   = wceq 1363   ≠ wne 2357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710

This theorem depends on definitions:  df-bi 117  df-ne 2358

This theorem is referenced by:  nelpri  3628  nelprd  3630  eldifpr  3631  0nelop  4260  lcmgcd  12092  lcmdvds  12093  lgsdirnn0  14744  lgsdinn0  14745