Theorem neorian 3020

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 2926	. . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
2		df-ne 2926	. . 3 ⊢ (𝐶 ≠ 𝐷 ↔ ¬ 𝐶 = 𝐷)
3	1, 2	orbi12i 914	. 2 ⊢ ((𝐴 ≠ 𝐵 ∨ 𝐶 ≠ 𝐷) ↔ (¬ 𝐴 = 𝐵 ∨ ¬ 𝐶 = 𝐷))
4		ianor 983	. 2 ⊢ (¬ (𝐴 = 𝐵 ∧ 𝐶 = 𝐷) ↔ (¬ 𝐴 = 𝐵 ∨ ¬ 𝐶 = 𝐷))
5	3, 4	bitr4i 278	1 ⊢ ((𝐴 ≠ 𝐵 ∨ 𝐶 ≠ 𝐷) ↔ ¬ (𝐴 = 𝐵 ∧ 𝐶 = 𝐷))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ≠ wne 2925

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 207  df-an 396  df-or 848  df-ne 2926

This theorem is referenced by:  neneor  3025  poxp2  8122  oeoa  8561  recextlem2  11809  crne0  12179  crreczi  14193  gcdcllem3  16471  bezoutlem2  16510  nrhmzr  20446  dsmmacl  21650  mhpmulcl  22036  txhaus  23534  itg1addlem2  25598  coeaddlem  26154  dcubic  26756  creq0  32659  sibfof  34331  rrx2pnecoorneor  48704