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Mirrors > Home > ILE Home > Th. List > neanior | GIF version |
Description: A De Morgan's law for inequality. (Contributed by NM, 18-May-2007.) |
Ref | Expression |
---|---|
neanior | ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐷) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐶 = 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ne 2341 | . . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
2 | df-ne 2341 | . . 3 ⊢ (𝐶 ≠ 𝐷 ↔ ¬ 𝐶 = 𝐷) | |
3 | 1, 2 | anbi12i 457 | . 2 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐷) ↔ (¬ 𝐴 = 𝐵 ∧ ¬ 𝐶 = 𝐷)) |
4 | pm4.56 775 | . 2 ⊢ ((¬ 𝐴 = 𝐵 ∧ ¬ 𝐶 = 𝐷) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐶 = 𝐷)) | |
5 | 3, 4 | bitri 183 | 1 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐷) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐶 = 𝐷)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 ↔ wb 104 ∨ wo 703 = 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-ne 2341 |
This theorem is referenced by: nelpri 3605 nelprd 3607 eldifpr 3608 0nelop 4231 lcmgcd 12021 lcmdvds 12022 lgsdirnn0 13703 lgsdinn0 13704 |
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