Theorem neorian 3038

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 2942	. . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
2		df-ne 2942	. . 3 ⊢ (𝐶 ≠ 𝐷 ↔ ¬ 𝐶 = 𝐷)
3	1, 2	orbi12i 914	. 2 ⊢ ((𝐴 ≠ 𝐵 ∨ 𝐶 ≠ 𝐷) ↔ (¬ 𝐴 = 𝐵 ∨ ¬ 𝐶 = 𝐷))
4		ianor 981	. 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 205   ∧ wa 397   ∨ wo 846   = wceq 1542   ≠ wne 2941

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 206  df-an 398  df-or 847  df-ne 2942

This theorem is referenced by:  neneor  3043  poxp2  8129  oeoa  8597  recextlem2  11845  crne0  12205  crreczi  14191  gcdcllem3  16442  bezoutlem2  16482  dsmmacl  21296  mhpmulcl  21692  txhaus  23151  itg1addlem2  25214  coeaddlem  25763  dcubic  26351  creq0  31960  sibfof  33339  nrhmzr  46647  rrx2pnecoorneor  47401