Theorem ne3anior 2428

Description: A De Morgan's law for inequality. (Contributed by NM, 30-Sep-2013.) (Proof rewritten by Jim Kingdon, 19-May-2018.)

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		df-ne 2341	. . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
2		df-ne 2341	. . 3 ⊢ (𝐶 ≠ 𝐷 ↔ ¬ 𝐶 = 𝐷)
3		df-ne 2341	. . 3 ⊢ (𝐸 ≠ 𝐹 ↔ ¬ 𝐸 = 𝐹)
4	1, 2, 3	3anbi123i 1183	. 2 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐷 ∧ 𝐸 ≠ 𝐹) ↔ (¬ 𝐴 = 𝐵 ∧ ¬ 𝐶 = 𝐷 ∧ ¬ 𝐸 = 𝐹))
5		3ioran 988	. 2 ⊢ (¬ (𝐴 = 𝐵 ∨ 𝐶 = 𝐷 ∨ 𝐸 = 𝐹) ↔ (¬ 𝐴 = 𝐵 ∧ ¬ 𝐶 = 𝐷 ∧ ¬ 𝐸 = 𝐹))
6	4, 5	bitr4i 186	1 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐷 ∧ 𝐸 ≠ 𝐹) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐶 = 𝐷 ∨ 𝐸 = 𝐹))

Syntax hints:  ¬ wn 3   ↔ wb 104   ∨ w3o 972   ∧ w3a 973   = wceq 1348   ≠ wne 2340

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-ne 2341

This theorem is referenced by:  eldiftp  3629