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 2944 | . . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
2 | df-ne 2944 | . . 3 ⊢ (𝐶 ≠ 𝐷 ↔ ¬ 𝐶 = 𝐷) | |
3 | 1, 2 | orbi12i 912 | . 2 ⊢ ((𝐴 ≠ 𝐵 ∨ 𝐶 ≠ 𝐷) ↔ (¬ 𝐴 = 𝐵 ∨ ¬ 𝐶 = 𝐷)) |
4 | ianor 979 | . 2 ⊢ (¬ (𝐴 = 𝐵 ∧ 𝐶 = 𝐷) ↔ (¬ 𝐴 = 𝐵 ∨ ¬ 𝐶 = 𝐷)) | |
5 | 3, 4 | bitr4i 277 | 1 ⊢ ((𝐴 ≠ 𝐵 ∨ 𝐶 ≠ 𝐷) ↔ ¬ (𝐴 = 𝐵 ∧ 𝐶 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 396 ∨ wo 844 = wceq 1539 ≠ wne 2943 |
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 397 df-or 845 df-ne 2944 |
This theorem is referenced by: neneor 3044 oeoa 8428 recextlem2 11606 crne0 11966 crreczi 13943 gcdcllem3 16208 bezoutlem2 16248 dsmmacl 20948 mhpmulcl 21339 txhaus 22798 itg1addlem2 24861 coeaddlem 25410 dcubic 25996 creq0 31070 sibfof 32307 poxp2 33790 nrhmzr 45431 rrx2pnecoorneor 46061 |
Copyright terms: Public domain | W3C validator |