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Mirrors > Home > ILE Home > Th. List > ne3anior | GIF version |
Description: A De Morgan's law for inequality. (Contributed by NM, 30-Sep-2013.) (Proof rewritten by Jim Kingdon, 19-May-2018.) |
Ref | Expression |
---|---|
ne3anior | ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐷 ∧ 𝐸 ≠ 𝐹) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐶 = 𝐷 ∨ 𝐸 = 𝐹)) |
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)) |
Colors of variables: wff set class |
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 3627 |
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