Theorem neorian 2888

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 2795	. . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
2		df-ne 2795	. . 3 ⊢ (𝐶 ≠ 𝐷 ↔ ¬ 𝐶 = 𝐷)
3	1, 2	orbi12i 543	. 2 ⊢ ((𝐴 ≠ 𝐵 ∨ 𝐶 ≠ 𝐷) ↔ (¬ 𝐴 = 𝐵 ∨ ¬ 𝐶 = 𝐷))
4		ianor 509	. 2 ⊢ (¬ (𝐴 = 𝐵 ∧ 𝐶 = 𝐷) ↔ (¬ 𝐴 = 𝐵 ∨ ¬ 𝐶 = 𝐷))
5	3, 4	bitr4i 267	1 ⊢ ((𝐴 ≠ 𝐵 ∨ 𝐶 ≠ 𝐷) ↔ ¬ (𝐴 = 𝐵 ∧ 𝐶 = 𝐷))

Syntax hints:  ¬ wn 3   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1483   ≠ wne 2794

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 197  df-or 385  df-an 386  df-ne 2795

This theorem is referenced by:  neneor  2893  oeoa  7677  wemapso2lem  8457  recextlem2  10658  crne0  11013  crreczi  12989  gcdcllem3  15223  bezoutlem2  15257  dsmmacl  20085  txhaus  21450  itg1addlem2  23464  coeaddlem  24005  dcubic  24573  sibfof  30402  nrhmzr  41873