![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > neorian | Structured version Visualization version GIF version |
Description: A De Morgan's law for inequality. (Contributed by NM, 18-May-2007.) |
Ref | Expression |
---|---|
neorian | ⊢ ((𝐴 ≠ 𝐵 ∨ 𝐶 ≠ 𝐷) ↔ ¬ (𝐴 = 𝐵 ∧ 𝐶 = 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ne 2933 | . . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
2 | df-ne 2933 | . . 3 ⊢ (𝐶 ≠ 𝐷 ↔ ¬ 𝐶 = 𝐷) | |
3 | 1, 2 | orbi12i 544 | . 2 ⊢ ((𝐴 ≠ 𝐵 ∨ 𝐶 ≠ 𝐷) ↔ (¬ 𝐴 = 𝐵 ∨ ¬ 𝐶 = 𝐷)) |
4 | ianor 510 | . 2 ⊢ (¬ (𝐴 = 𝐵 ∧ 𝐶 = 𝐷) ↔ (¬ 𝐴 = 𝐵 ∨ ¬ 𝐶 = 𝐷)) | |
5 | 3, 4 | bitr4i 267 | 1 ⊢ ((𝐴 ≠ 𝐵 ∨ 𝐶 ≠ 𝐷) ↔ ¬ (𝐴 = 𝐵 ∧ 𝐶 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 ∨ wo 382 ∧ wa 383 = wceq 1632 ≠ wne 2932 |
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 384 df-an 385 df-ne 2933 |
This theorem is referenced by: neneor 3031 oeoa 7846 wemapso2lem 8622 recextlem2 10850 crne0 11205 crreczi 13183 gcdcllem3 15425 bezoutlem2 15459 dsmmacl 20287 txhaus 21652 itg1addlem2 23663 coeaddlem 24204 dcubic 24772 sibfof 30711 nrhmzr 42383 |
Copyright terms: Public domain | W3C validator |